“While President Biden and Democrats work to lower costs and continue the historic economic recovery made possible by the American Rescue Plan, Republicans have done everything they can to try to stand in the way,” Jaime Harrison, the chairman of the Democratic National Committee, said in a statement.

Yet months of national polls show that Americans have a vastly different perception of the party in power. Even in overwhelmingly liberal Los Angeles, private Democratic polling in April found Mr. Biden’s favorability rating at only 58 percent, according to a person with direct knowledge of the data.

Democratic tensions over messaging have been on display in Ohio, where candidates in this week’s primaries reflect the full spectrum of competing views.

Ms. Brown, who faces a contested primary in a safely Democratic seat and was endorsed by Mr. Biden, is running hard on the bipartisan infrastructure law.

She echoed other House Democrats in promoting the message that “Democrats have been delivering.”

But Biden advisers have privately indicated that pitch tests poorly as a party slogan. And at another Ohio event in late April, Nina Turner, a former state senator who is challenging Ms. Brown from the left in a rematch, suggested that Democrats had not delivered nearly enough.

She urged, among other priorities, universal cancellation of student debt — or, at a minimum, canceling $10,000 in federal student debt per borrower (Ms. Brown also supports some student debt forgiveness measures). Mr. Biden, who endorsed the $10,000 goal in 2020, has postponed payments, and significant student debt has been erased during his tenure, but some have called on him to do much more. He may take further action, and there is still time to make more progress on the Democratic agenda.

But for now, many on the left are disappointed that Democrats, despite controlling Washington, have run aground in the divided Senate on priorities like the climate and voting rights.