The former Florida governor took advantage of a policy-focused debate to detail positions on the economy and immigration.
The televised debate is the last for the Republicans until mid-December, and could help shape the course of the campaign into the winter.
Voters are beginning to pay more attention to the White House race less than three months before the lead-off Iowa caucuses start the state-by-state nominating contests.
Billionaire Donald Trump’s grip on the Republican field has been challenged in recent weeks by retired neurosurgeon Ben Carson, another outsider.
As Carson has risen in the polls, however, he has faced a flurry of questions about his biography, which has been central to his connection with voters.
He sought to steady his campaign in the debate, swatting away questions about the veracity of his celebrated biography detailing his rise from poverty to become a neurosurgeon.
“We should vet all candidates,” Carson said. “I have no problem with being vetted. What I do have a problem with is being lied about.”
While pieces of his background had been challenged earlier in the campaign, the questions ballooned last week.
CNN reported it could not find friends or confidants to corroborate the story, told in his autobiography, of his unsuccessfully trying to stab a close friend when he was a teenager.
Later in the week, Politico examined Carson’s claim of having been offered a scholarship to attend the US Military Academy. And the Wall Street Journal said it could not confirm anecdotes told by Carson about his high school and college years.
The debate in Milwaukee opened with a narrow focus on economic policy, with moderators from Fox Business News allowing the candidates to deliver lengthy answers and avoiding attempts to get them to engage with one another.
Bush entered the debate in need of a strong performance to ease the anxiety of donors and other supporters.
He sharply criticised Clinton for defending President Barack Obama’s economic policies.
Bush also stood by his call for immigration policies that would allow some of those in the US illegally to find a path to legal status. He criticised Trump’s call for mass deportations as an impractical plan that would hand Democrats a talking point as they seek to appeal to Hispanic voters.
“They’re doing high fives in the Clinton campaign when they’re hearing this,“ Bush said.
He avoided tangling with fellow Floridian Marco Rubio, a shift in strategy from his poor performance in the last debate.