US President Joe Biden has hit back at criticism over his age, telling supporters in a fiery speech that he will win re-election in November after a poor debate performance fuelled concern about his candidacy.

"I know I'm not a young man, to state the obvious," he told a rally in the battleground state of North Carolina on Friday, one day after he struggled in the televised showdown with his Republican rival Donald Trump.

"I don’t walk as easy as I used to... I don’t debate as well as I used to," he acknowledged. "But I know what I do know, I know how to tell the truth [and] I know how to do this job."

Mr Biden, 81, said he believed with his "heart and soul" that he could serve another term, as the cheering crowd in Raleigh chanted “four more years”.

Trump, meanwhile, held a rally of his own in Virginia just hours later, where he hailed a "big victory" in the debate, which CNN said was viewed by 48 million people on television and millions more online. "Joe Biden's problem is not his age," the 78-year-old Trump said. "It's his competence. He's grossly incompetent."

The former president said he did not believe speculation that Mr Biden would drop out of the race, saying he "does better in polls" than other Democrats, including California Governor Gavin Newsom and Vice-President Kamala Harris.

While questions over Mr Biden's age are not new, his shaky performance on the debate stage - which was marked by verbal blanks, a hoarse voice and some difficult-to-follow answers - triggered panic among some Democrats who raised fresh questions about his candidacy.

Democratic officials, political operatives, and people close to the president who spoke to the BBC's Katty Kay painted a picture of an anxious party concerned about the strength of their candidate.

Nancy Pelosi, the former Democratic House speaker, said that “from a performance standpoint it wasn’t great”. Other Democrats, such as Biden’s former communications director Kate Bedingfield, called it “a really disappointing debate performance”.

Democratic donors who spoke anonymously to various media outlets were more forthright, with one calling the performance "disqualifying". "The only way it could have been more disastrous was if he had fallen off the stage. Big donors are saying... he has to go,” one Democratic operative told the Financial Times.

And on Friday, the New York Times editorial board called on Mr Biden to drop out. It said Democrats should "acknowledge that Mr Biden can’t continue his race, and create a process to select someone more capable to stand in his place".

Despite concern from some media pundits, early indications suggest there has been "no change" in the polls in the aftermath of the debate, the Washington Post columnist Philip Bump told BBC R4's Today programme.

However, Mr Bump pointed out no "high-quality" polls have been conducted since the debate aired.

Speaking later on the programme, pollster Frank Luntz said most Americans have already made up their minds about who they will vote for later this year.

But publicly, many senior Democrats and Biden allies defended his performance as they sought to calm liberal jitters on Friday. Among those to rally behind Mr Biden were former President Barack Obama, who tweeted that "bad debate nights happen".

"This election is still a choice between someone who fought for ordinary folks his entire life and someone who only cares about himself," Mr Obama wrote, adding that Mr Trump is "someone who lies through his teeth for his own benefit".

Mr Biden and his campaign were quick to dismiss calls for him to step down as the candidate.

"President Biden is the only person who has ever beaten Donald Trump. He will do it again," a campaign adviser said. “This election was never going to be won or lost in one rally, one conversation, or one debate. "

The Biden campaign also said the president had raised $14m from fundraisers in recent days, in an apparent effort to show it was maintaining momentum.

Mr Biden is expected to meet donors on Friday and Saturday, including at events in Manhattan and the wealthy Hamptons.