By Doina Chiacu

WASHINGTON (Reuters) - Republican presidential candidate Chris Christie has dismissed him as a puppet and former mentor Jeb Bush called him not fit to serve as up-and-coming Republican Marco Rubio becomes the man to beat in next week's New Hampshire primary.

Rubio's unexpectedly strong third-place showing in Monday's Iowa caucuses has made him the target of rival Republican candidates who focused their campaigns on the New England state's first-in-the-nation primary.

Christie, the New Jersey governor, former Florida Governor Jeb Bush and Ohio Governor John Kasich all hope for a strong showing in New Hampshire to boost their flagging campaigns to become the Republican nominee in the Nov. 8 election to replace Democratic President Barack Obama.

While Christie said on Thursday he would not be out of the race if he lost to Rubio in New Hampshire, Kasich has told audiences he would go back to Ohio if he got "smoked" there.

U.S. Senator Ted Cruz, who won the Iowa caucuses on strong evangelical Christian credentials, was expected to have less appeal for voters in New Hampshire. Real estate mogul Donald Trump's second-place showing in Iowa raised questions about how well his popularity can survive the voting booth.

Christie has been one of Rubio's fiercest critics this week, calling the first-term U.S. senator from Florida "the boy in the bubble" the day after the Iowa vote. He continued the personal vein of attack on Thursday.

"This isn't the most controlled candidate we've seen in this race at all. His handlers handle him all the time," Christie said on Fox News.

"We need to take him out of that controlled atmosphere because, believe me, it won't be controlled against Hillary Clinton this fall," he said.

Clinton, the former secretary of state, is battling U.S. Senator Bernie Sanders of Vermont for the Democratic nomination.

Bush, whose establishment Republican credentials have not guaranteed him public support, has also turned on his former protege.

On Wednesday he took out a full-page ad in a leading New Hampshire newspaper, the Union-Leader, attacking Rubio as not ready to serve as commander in chief.

"Nearly every political leader in Florida of stature who knows Marco and Jeb's records have joined us in endorsing Governor Bush," the ad read.

Rubio has garnered significant endorsements since Monday, including U.S. Senator Tim Scott of South Carolina, which holds its primary this month, and U.S. Representative Matt Salmon of Arizona, a co-founder of the conservative House Freedom Caucus.

(Additional reporting by Susan Heavey; Editing by Andrea Ricci)