In California’s 45th Congressional District, along Western Avenue in Buena Park, a giant billboard is set to display a photograph of Representative Michelle Steel next to former President Donald J. Trump and Representative Jim Jordan, the Republican hard-liner from Ohio she voted for twice this week for speaker.
“Rep. Steel Supports Extremism,” the billboard reads. “Stop the extremism.”
The advertising campaign, paid for by the Progressive Change Campaign Committee, is part of a broad effort by Democrats to target Republicans like Ms. Steel, who represent congressional districts that President Biden won in 2020. A dozen of those vulnerable G.O.P. lawmakers have stood on the House floor this week and cast their votes to put Mr. Jordan second in line to the presidency.
Another group, the Congressional Integrity Project, began a digital ad campaign this week in those same districts, focusing on Mr. Jordan and his attempts to overthrow the 2020 election.
“Every House Republican who votes for Jim Jordan to be speaker of the House should be held accountable for his efforts to overturn the 2020 election, his role in the Jan. 6 fake electors plot, and his continued attacks on our democracy,” said Kyle Herrig, the executive director of the advocacy organization.
The latest round of House Republican infighting has badly damaged the G.O.P. brand. It has left the party leaderless and one chamber of Congress paralyzed for more than two weeks. The chaos is raising the chances that Democrats could win back the majority next year, and it has given them ample ammunition for their campaign narrative, which casts Republicans as right-wing extremists who are unfit to govern.
“It hurts the country; it hurts the Congress; it’s hurting our party,” said Representative Don Bacon of Nebraska, one of 18 Republicans who represent districts won by Mr. Biden in 2020. “It’s putting us in a bad hole for next November.”
He said his hard-right colleagues who moved to oust Speaker Kevin McCarthy earlier this month and touched off the intractable scramble to replace him “want to be in the minority.”
Mr. Bacon added: “I think they would prefer that. So they can just vote no and just yell and scream all the time.”
The Nebraska lawmaker opposed Mr. Jordan’s candidacy, but he and other mainstream G.O.P. lawmakers worry that, no matter who is ultimately elected speaker, the Ohio Republican’s nomination has only boosted Democrats’ efforts to tie them to the most hard-right members of their party, placing their seats at risk in 2024.
Republican groups are pushing back on a narrative of extremism and dysfunction. The American Action Network is running an ad campaign lauding 16 Republicans in Biden districts who voted to prevent a government shutdown last month.
But Democrats have seized every opportunity to condemn Mr. Jordan, and Republicans for backing him.
“Jim Jordan is the poster boy for MAGA extremism,” Representative Hakeem Jeffries of New York, the minority leader, said on Tuesday night.
In a speech on the House floor in which he nominated Mr. Jeffries for speaker, Representative Pete Aguilar of California, the No. 3 House Democrat, laid out a case against Mr. Jordan that could have doubled as a template for a campaign attack ad against any Republican who supported him.
“A vote today to make the architect of a nationwide abortion ban, a vocal election denier and an insurrection inciter to the speaker of this House would be a terrible message to the country and our allies,” Mr. Aguilar said.
The candidacy of Mr. Jordan, the combative co-founder of the ultraconservative Freedom Caucus and a key player in Mr. Trump’s efforts to overturn the 2020 election results, has left many House Republicans in a no-win position.
If Mr. Jordan were to prevail — a prospect that appeared less likely on Wednesday after he lost a second ballot — his ascension would cement the House G.O.P.’s reputation as an extreme group that is composed of Mr. Trump’s loyal foot soldiers. And if the hard-right lawmaker fails, the disarray could in the House drag on, only hardening the view of Republicans as completely incapable of governing.
For mainstream Republicans representing politically competitive districts, the damage may already be done regardless of the outcome of the vote, or how many rounds it takes.
“It’s hard to present yourself as a figure of bipartisan compromise and moderation when you vote for someone who resolutely stands against any bipartisan compromise and is the furthest thing from a moderate a voter can imagine,” said Geoff Garin, a Democratic pollster.
Democrats have been circulating data from YouGov Blue, a research division for progressive and Democratic clients, that found that 63 percent of respondents in a recent poll said that moderate Republicans should work with Democrats to form a bipartisan governing coalition. Only 37 percent of respondents said those moderate Republicans should only work with other Republicans to elect a new speaker.
Christina Bohannan, a Democrat challenging Representative Mariannette Miller-Meeks in a competitive Iowa district, said her opponent’s vote for Mr. Jordan on Tuesday “shows her true values and how contrary they are to Iowa values.”
She said that more than 60 percent of Iowans reject Mr. Jordan’s position in support of a nationwide abortion ban, and noted that he has never voted for a farm bill, one of the most critical issues for the state.
“This is a real slap in the face to Iowa women for Miller-Meeks to support him,” Ms. Bohannan said. “I can’t imagine a clearer example of how Representative Miller-Meeks is selling out Iowans to the extreme members of her party rather than take a more bipartisan position.”
Ms. Miller-Meeks on Wednesday switched her vote, backing off her support for Mr. Jordan and voting for Representative Kay Granger of Texas, instead. But the damage had already been done.
Some Republicans were admitting as much on Tuesday.
“There’s no way we win the majority if the message we send to the American people is we believe in the election was stolen, and we believe that Jan. 6 was a tour of the Capitol,” Representative Ken Buck, Republican of Colorado, said on CNN. Mr. Buck was one of 22 Republicans who refused to vote for Mr. Jordan on Wednesday. He has said he is opposed in part because Mr. Jordan has been unwilling to say that Mr. Biden won the 2020 election.
Representative Jen Kiggans, who represents a Virginia district won by Mr. Biden, also was outspoken about her opposition to Mr. Jordan.
“Mr. Jordan’s government funding plan has the potential to further cut the defense budget, which is already inadequate,” she said in a video posted on social media. She also voiced concerns about his vote last month against keeping the government open.
Bullish Republicans dismissed the chaos that has ground the House to a standstill as “Beltway drama” that would be forgotten by November next year.
“When the dust settles, not a single competitive House race has changed,” said Dan Conston, the president of the Congressional Leadership Fund, the main House Republican super PAC. “Our targeted incumbents are strong and have cultivated individual brands.” He added that “our recruits are far superior” to those of the Democrats.
Republicans have also noted that the political climate overall remained positive for them.
A recent ABC poll, for instance, showed Mr. Biden with a 26 percent approval rating on immigration and border security; a 29 percent approval rating on inflation; and a 33 percent approval rating on crime.
A poll conducted last month by Gallup showed 57 percent of respondents saying Republicans were better at protecting the country from international terrorism and military threats, compared to 35 percent for Democrats — the widest gap Gallup has registered since it began polling the question in 2002.
Still, the political climate isn’t substantially different from last year, when Republicans failed to achieve the expected sweeping victories and won the House majority by just four seats. One of the only things that has shifted since then is the demonstrable failure of House Republicans to govern.
Strategists noted that even if the speaker fight had never happened, mainstream Republicans already were facing a difficult political challenge with Mr. Trump the most likely presidential candidate to top their party’s ticket.
Some Republicans from swing districts said this week that they hoped their voters would be able to separate them from the chaos surrounding their party.
“I am going to be judged by the work that I do, and whoever runs against me is going to be judged based on their experience and livelihood,” said Representative Marc Molinaro of New York, who voted for Mr. Jordan and represents a district Mr. Biden won in 2020. “I truly believe that at the end of the day, if we’re earnest and honest with the people we represent and authentic in that service, they’ll judge us based on that.”
Luke Broadwater and Kayla Guo contributed reporting.
