By Joshua Jamerson, Sabrina Siddiqui, Wall Street Journal–

Chaos at Tuesday’s fiery presidential debate prompted organizers to promise to change the structure of the remaining matchups between President Trump and Democrat Joe Biden, as members of both parties criticized the president on Wednesday for his deflections when asked to condemn white-supremacist groups.

Politicians from both parties on Wednesday said the debate a day earlier in Cleveland wasn’t informative, with both Messrs. Trump and Biden at times talking over each other and lobbing insults, though the president appeared to interrupt his opponent more often.

The nonpartisan Commission on Presidential Debates, which sponsors televised debates and selects the moderators, said that it soon would announce changes “to ensure a more orderly discussion of the issues.” These could include tweaking how much time a candidate could speak and whether the moderator could shut off a candidate’s microphone, according to one member of the commission’s board.

While the commission said moderator Chris Wallace of Fox News showed skill in hosting Tuesday’s debate, Mr. Wallace repeatedly had to step in and frequently chided the Republican president for interrupting his challenger.

The Trump campaign blasted the move. “They shouldn’t be moving the goal posts and changing the rules in the middle of the game,” spokesman Tim Murtaugh said.

Mr. Biden’s campaign didn’t respond to a request for comment on the announcement. On the campaign trail earlier in the day, the former vice president said he hoped that in future debates, “there’s a way in which the debate commission can control the ability of us to answer the question without interruption.”

Tuesday’s first of three debates offered the president’s campaign one of its last major opportunities to change the dynamics of a race that polls show he is trailing nationally and in key states. About 73.1 million people watched the debate on TV, according to Nielsen, down 13% from the first debate of the 2016 cycle.

While Mr. Trump said Wednesday he had received good reviews for his performance, his comments about white nationalists reverberated.

Mr. Wallace at one point asked Mr. Trump if he would condemn white supremacist groups and urge them to stand down in clashes such as those in Kenosha, Wis., and Portland, Ore. Mr. Trump said: “Sure, I’m willing to do that, but I would say, I would say almost everything I see is from the left wing.”

After Mr. Biden urged the president to denounce the far-right group Proud Boys, Mr. Trump said: “Proud Boys. Stand back, and stand by, but I’ll tell you what, I’ll tell you what: Somebody’s got to do something about antifa and the left.”

Mr. Biden, who was on a train tour Wednesday aimed at highlighting his plans to revitalize U.S. manufacturing, provided his own message to the Proud Boys, a group that engages in political violence, would be to “cease and desist.”

Mr. Trump, asked about the group Wednesday, said, “I don’t know who the Proud Boys are. Whoever they are, they need to stand down.”

Rep. Kevin McCarthy of California, the House GOP leader, defended Mr. Trump. “I think the president’s been very clear that he’s against anybody for committing violence in the streets,” Mr. McCarthy said. “How many times does he have to say it? If the question is would you denounce it and the answer is yes, he did that.”

However, some fellow Republicans said he should have taken a clearer stance against such groups.

Sen. Susan Collins (R., Maine), a centrist who is in a tough re-election battle against a well-financed Democratic challenger, said Mr. Trump shouldn’t have been so vague on the matter. “That was a mistake,” Ms. Collins said. Ms. Collins also called the matchup “the least educational debate of any presidential debate I’ve ever seen.”

Sen. Tim Scott of South Carolina, one of three Black members of the Senate and the only one who is a Republican, said he thought the president misspoke when he deflected the white-supremacist question. “I think he should correct it. If he doesn’t correct it, I guess he didn’t misspeak,” said Mr. Scott, who in the past has been critical of the president on issues of race.

The other presidential debates are scheduled for Oct. 15 and 22, with a vice-presidential debate set for next week.

Senate Majority Leader Mitch McConnell (R., Ky.) said he agreed with Mr. Scott. “With regard to the white supremacy issue, I want to associate myself with the remarks of Sen. Tim Scott that he put out earlier today, exactly correctly, and that’s exactly how I would express myself on this issue,” Mr. McConnell said. “He said it was unacceptable not to condemn white supremacy. And so I do so in the strongest possible way.” Mr. McConnell also said that Mr. Trump’s remarks shouldn’t hurt the campaigns of GOP senators like Ms. Collins. “I don’t know any of my colleagues who will have problems as a result of that,” he said.

Newton Minow, a longtime member of the debate commission who sits on its board of directors, said he couldn’t remember another time a debate sponsor sought to change the rules once they had already been agreed upon by both campaigns. “This is unprecedented,” he said.

“Everything has been discussed,” Mr. Minow said of potential tweaks. He declined to go into further details but said the commission was expected to make public changes soon.

On Tuesday night, Messrs. Trump and Biden also touched on the Supreme Court vacancy, the administration’s handling of the coronavirus and more. Mr. Trump targeted his Democratic rival’s son for criticism, while Mr. Biden said the other man was “the worst president that America has ever had.”

At one point in the debate, Mr. Trump said he had the support of law-enforcement groups around the country including in Portland, Ore. “The sheriff there just came out today and said, ‘I support President Trump,’” Mr. Trump said. The sheriff, Mike Reese, responded in a tweet: “As the Multnomah County Sheriff I have never supported Donald Trump and will never support him.’’

Mr. Trump sought to remind voters in a tweet Wednesday that Mr. Biden has tried to navigate a tricky line in his party between moderates and liberals. The president and his allies also pointed to Mr. Biden’s refusal to say whether he supports adding more justices to the Supreme Court, which some Democrats are pushing after the death of liberal Justice Ruth Bader Ginsburg.

“I’m not going to answer the question,” Mr. Biden said during the debate when prodded by Mr. Trump.

Earlier this month, when asked whether Mr. Biden’s prior opposition to adding seats stood in the wake of Justice Ginsburg’s death, a Biden spokesman declined to answer but added that Mr. Biden in a recent speech said “we need to de-escalate—not escalate.” Democrats said the debate was a sad moment in American history. “Last night we witnessed a debacle as it relates to our democracy,” Rep. Hakeem Jeffries (D., N.Y.), a top House Democratic leader, said.

Rep. Tim Ryan (D., Ohio), who attended the debate in his home state, called the evening “the worst 90 minutes of my adult life” and said it wouldn’t sway many suburban voters, who are an important constituency this fall. “I don’t know one suburban mom that I’ve ever met in my 47 years that would watch what happened last night and would go, ‘Oh, I’m going to go for him,’ ” Mr. Ryan said, referring to Mr. Trump. “He came across last night like every woman’s ex-husband.”

Write to Joshua Jamerson at joshua.jamerson@wsj.com and Sabrina Siddiqui at Sabrina.Siddiqui@wsj.com