Republican presidential candidate Jeb Bush shifted into attack mode Thursday, railing against front-runner Donald Trump and accusing his billionaire rival of flooding the race with a “tidal wave of accusations [and] bombastic talk.”

Firing back after a string of insults and dismissive remarks from the real estate tycoon, the former Florida governor accused Mr. Trump of being a vanity candidate, offering “outrageous” ideas and being a fake conservative.

“All of this stuff [from Mr. Trump] is to appeal to people’s anger and their angst, rather than have solutions to solve problems,” Mr. Bush told reporters after a town hall meeting.

Mr. Bush is taking an aggressive stance as Mr. Trump draws huge crowds in the early voting state of New Hampshire and elsewhere on the campaign trail. Mr. Trump also leads in every recent poll, including one that showed him on top in Mr. Bush’s home state of Florida.

Mr. Trump leads the crowded field of Republican candidates with 21 percent support among likely GOP primary voters in Florida, followed by Mr. Bush with 17 percent, according to a Quinnipiac University poll.

In New Hampshire, a Boston Herald/Franklin Pierce University poll last week showed Mr. Trump leading with 18 percent, followed by Mr. Bush with 13 percent and Ohio Gov. John Kasich with 12 percent.

Mr. Bush insists that Mr. Trump’s lead will not last.

“You don’t win when you’re the large dog in the room, where it’s all about you. You win when you connect with people about their aspirations, not about how great you are, how rich you are,” Mr. Bush said at a town hall meeting in Keene.

“That’s not leadership,” he said.

Later, when a reporter asked about the groundswell of support for Mr. Trump, Mr. Bush declared: “There is a big difference between Donald Trump and me.”

He then fired off a litany of reasons that he said made him, not Mr. Trump, the true conservative leader in the race.

“I’m a proven conservative with a record. He isn’t. I cut taxes every year. He’s proposed the largest tax increase in mankind’s history, not just our own country’s history,” Mr. Bush said.

He slammed Mr. Trump for having supported massive tax increases, partial-birth abortion and a government-financed single-payer health care system.

“I’ve been consistently pro-life. He said he was for partial-birth abortion. I’ve never met a person that actually thought that was a good idea,” Mr. Bush said.

“He’s been a Democrat longer than he’s been a Republican,” he said. “I have fought for Republican and conservative causes all of my adult life, and I just think when people get this narrative … then they are going to find that I’m going to be the guy that they are going to vote for.”

Mr. Bush noted that Mr. Trump’s immigration plan includes building a wall along the entire border with Mexico and deporting every illegal immigrant.

He said the proposals were “outrageous and just don’t make sense.”

“The immigration policies that he has brought up are not conservative either,” said Mr. Bush. “It’s going to cost hundreds and hundreds of billions of dollars.”

Mr. Trump has taken swipes at Mr. Bush repeatedly.

At a town hall meeting Wednesday in Derry, New Hampshire, Mr. Trump mocked Mr. Bush for saying that the U.S. needs to show Iraq that it has some “skin in the game.”

“After years of fighting,” said Mr. Trump, “can you believe — ‘skin in the game?’ And I thought that was incredibly dumb.”

He said the comment was as dumb as Mr. Bush’s description of illegal immigration as an “act of love.”

“I don’t see how he’s electable,” said Mr. Trump. “Jeb Bush is a low-energy person. For him to get things done is hard. He’s very low-energy.”

He joked about the smaller crowds at Mr. Bush’s gatherings and said those in attendance were “sleeping now” that Mr. Bush was speaking.

Outside the town hall meeting in Keene, Mr. Bush said the race would be “a long haul” and vowed to overtake Mr. Trump.

“At the end of the day, the candidate who offers the most compelling solutions to the problems that we face, who has the proven leadership skills to do it, is going to be the one that wins,” Mr. Bush said.