Ukrainian forces also said they pressed an offensive south and east of the southern port of Mykolayiv, moving in the direction of Kherson, the only Ukrainian regional capital occupied by Russia since the war began Feb. 24. Ukraine said it carried out an airstrike on the Kherson airport, which is now a Russian air base, and satellite imagery of the tarmac showed seven destroyed or damaged Russian helicopters, some of them engulfed in flames. Kyiv also said it shot down two Russian Su-30SM jets over the Black Sea off Odessa.

Moscow has largely not commented on combat losses and said its campaign was progressing.

Mr. Zelensky, in a virtual address to American lawmakers on Wednesday, thanked the U.S. for its support as Ukraine tries to fend off Russia’s invasion, but he added, “In the darkest time for our country, for the whole of Europe, I call for you to do more.”

He renewed his appeal for a no-fly zone and added that, absent such a move—which NATO has ruled out—more weapons should be provided. He also called for the imposition of further sanctions on Russia. And he asked members of Congress to pressure American companies in their district that are still doing business in Russia to get them to stop.

Mr. Zelensky’s appeal came as civilian casualties mounted from Russia’s air and artillery strikes. In the western city of Rivne, officials raised the death toll from an attack on a television tower earlier this week, saying 21 people were killed, up from nine fatalities on the day of the strike. In the northern city of Chernihiv, an artillery shell landed near a bakery, killing at least 10 people standing in line to buy bread, according to Ukraine news services.

Russia aimed for a quick capture of Kyiv in the first days of the war, and its forces thrust deep into Ukrainian territory up to the city’s outskirts in the north, west and east. But that offensive stalled out amid poor planning and logistics, and unexpectedly stiff resistance from Ukrainian forces armed with Western-provided antitank and anti-aircraft missiles.

The counteroffensive by Ukrainian armed forces “radically changes the parties’ disposition,” tweeted Mykhailo Podolyak, a Ukrainian presidential adviser who is negotiating a possible cease-fire with Russia. The talks, via videoconference, would resume today, he added, saying that there is room for compromise despite “fundamental contradictions.”

The main area of progress between the two sides concerns addressing mutual security concerns, a Ukrainian official said. Ukraine, ready to recognize that membership of the North Atlantic Treaty Organization is unlikely anytime soon, is considering dropping that in exchange for binding security guarantees from its Western partners and Russia, the official said. That, however, is only one item on the agenda, and there is little scope for compromise on Russia’s demands that Kyiv recognize the 2014 annexation of Crimea by Moscow and the self-proclaimed independence of the Russian-backed statelets in Donetsk and Luhansk.

“I don’t see a breakthrough—at this stage both sides believe they can win the war,” the official said.

“Any war ends in agreement,” Mr. Zelensky said in an overnight address. “As I am told, the positions in the negotiations sound more realistic. However, time is still needed for the decisions to be in Ukraine’s interests.”

Russian Foreign Minister Sergei Lavrov on Wednesday said that negotiations with Ukraine aren’t easy, but there are signs of compromise. Mr. Lavrov, cited by Russian state news agency TASS, said that the two sides were close to agreeing on some issues and that he held out hope that this could include the topic of Ukraine’s neutral status.

The Kremlin said Wednesday that a neutral status similar to those of Sweden or Austria could be a model for Ukraine. Kyiv has rejected that idea, seeking clear security guarantees.

Aleksii Arestovich, an adviser to Mr. Zelensky, said in an interview with Ukrainian media Tuesday that fighting in Ukraine could stop “relatively soon—that is in a week or two” if negotiations are successful.

Mr. Arestovich said that Ukraine is insisting that Russia withdraw all its forces from the country, and will not recognize Russia’s annexation of Crimea or the breakaway regions in Ukraine’s east.

He said that Russian President Vladimir Putin could still back out of the invasion by declaring victory and telling Russians at home that he has effectively demilitarized Ukraine by destroying its military equipment. Ukraine’s aspirations to join NATO, he said, are not an impediment, because the alliance’s members do not want Ukraine to join.

President Biden signed into law on Tuesday a government spending bill that includes $13.6 billion in security, humanitarian and economic aid for Ukraine. Mr. Biden is expected to announce more than $1 billion in new military assistance to Ukraine’s government as early as Wednesday, according to U.S. officials.

Weapons already provided by the U.S. and others have been indispensable in allowing Ukrainian forces to repel Russian attacks and hold the line so far, according to Ukrainian officials. However, as Russian airstrikes and missiles devastate Ukrainian cities, Kyiv is pleading for more, particularly equipment that would shore up the country’s air defenses.

Russian invasion

Areas no longer controlled by Ukraine as of Friday

Direction of invasion forces

Controlled by or allied to Russia

Primary refugee crossing locations

Chernobyl

Not in operation

Ukraine territory, recognized by Putin as independent

Nuclear facilities

Khmelnitsky

Controlled by

separatists

Zaporizhzhia

South

Ukraine

Transnistria

Sea of Azov

Sevastopol

Russian invasion

Areas no longer controlled by Ukraine as of Friday

Direction of invasion forces

Controlled by or allied to Russia

Ukraine territory, recognized by Putin as independent

Primary refugee crossing locations

Chernobyl

Not in operation

Nuclear facilities

Khmelnitsky

Controlled by

separatists

Zaporizhzhia

South

Ukraine

Transnistria

Sea of Azov

Sevastopol

Russian invasion

Areas no longer controlled by Ukraine as of Friday

Direction of invasion forces

Controlled by or allied to Russia

Primary refugee crossing locations

Ukraine territory, recognized by Putin as independent

Nuclear facilities

Chernobyl

Not in operation

Khmelnitsky

Controlled by

separatists

Zaporizhzhia

South

Ukraine

Transnistria

Sea of Azov

Sevastopol

Russian invasion

Areas no longer controlled by Ukraine as of Friday

Direction of invasion forces

Controlled by or allied to Russia

Primary refugee crossing locations

Nuclear facilities

Ukraine territory, recognized by Putin as independent

Transnistria

Sevastopol

Russian invasion

Areas no longer controlled by Ukraine as of Friday

Direction of invasion forces

Controlled by or allied to Russia

Primary refugee crossing locations

Nuclear facilities

Ukraine territory, recognized by Putin as independent

Transnistria

Sevastopol

On Tuesday, leaders of Poland, the Czech Republic and Slovenia traveled to Kyiv and met with Mr. Zelensky in a show of support for Ukraine.

In an emotional televised speech Tuesday, Mr. Zelensky implored Canadian politicians to increase military support and help implement a no-fly zone over his country to stop Russian bombardments. He told the Canadian lawmakers that 97 children had died since the Russian military began airstrikes on Ukraine.

“Please close the sky,” he said in his address to Canada’s parliament. “You all need to do more to stop Russia and protect Ukraine and protect Europe. They are destroying everything.”

NATO has turned down requests for a no-fly zone, saying it would risk drawing the West into war with Russia.

Separately, Ukraine’s Defense Minister Oleksii Reznikov is due to participate in a meeting of NATO defense ministers in Brussels, where the alliance is expected to discuss additional military assistance to the embattled country, as well as a reinforcement of its eastern flank to ward off Russia.

That gathering is a precursor to a hastily called meeting of NATO leaders on March 24. The White House said Tuesday that Mr. Biden would head to Brussels to participate as the West wrestles with how to intensify pressure on Moscow while avoiding direct military conflict between NATO and Russia.

The group of Central European leaders visiting Kyiv—all from NATO states—planned to offer a broad package of support for Ukraine, the Polish government said. Poland’s Prime Minister Mateusz Morawiecki and Deputy Prime Minister Jarosław Kaczyński, Czech Prime Minister Petr Fiala and Prime Minister of Slovenia Janez Janša met with Mr. Zelensky and Prime Minister Denys Shmyhal as representatives of the European Council. The delegation traveled to Kyiv jointly by train and arrived in the Ukrainian capital Tuesday evening.

“Europe must guarantee Ukraine’s independence and ensure that it is ready to help in Ukraine’s reconstruction,” Mr. Morawiecki said ahead of the meeting with Mr. Zelensky.

“I am sure that with such friends, with such countries, with such neighbors and partners, we will really be able to win,” Mr. Zelensky told reporters after the meeting.

He added that the talks focused on security guarantees for Ukraine and strengthening sanctions against Russia, as well as the prospects of Ukraine’s membership of the European Union, according to Mr. Zelensky’s official website.

Since Russian troops began their assault on Ukraine on Feb. 24, they have seized territory in the south of Ukraine but been stopped short around Kyiv and elsewhere. Increasingly, its forces have resorted to bombing residential areas and civilian infrastructure in an effort to wear down Ukrainian resistance.

In the besieged port city of Mariupol, an adviser to the mayor said the city’s death toll was significantly higher than the 2,357 casualties officially confirmed, putting it closer to 20,000. “It is currently impossible to determine the exact number of victims,” said Petro Andryushchenko.

Corpses lie in the street, as heavy shelling prevents residents from burying them, he said. “Death is everywhere.”

Survivors are crammed into basements without food, reduced to drinking water from the drains, Mr. Andryushchenko said. Russian forces have also trapped several hundred people in a hospital in Mariupol, according to Pavlo Kyrylenko, head of the regional military administration in the eastern region of Donetsk.

A convoy of food, water and medicine destined for the city has been held up for several days by Russian forces near Berdyansk. Some 20,000 people managed to flee the city in their own vehicles on Tuesday, Mr. Zelensky said.

More than three million people have fled Ukraine since the war began, according to United Nations figures.

—Georgi Kantchev contributed to this article.

Write to Alan Cullison at alan.cullison@wsj.com, Isabel Coles at isabel.coles@wsj.com and Yaroslav Trofimov at yaroslav.trofimov@wsj.com