For weeks, Donald J. Trump has romped through Iowa and New Hampshire without breaking a sweat, muscling out rivals for the Republican nomination and soaking up adoration from crowds convinced he will be the next president of the United States.

But as Mr. Trump marches steadily toward his party’s nomination, a harsher reality awaits him.

Outside the soft bubble of Republican primaries, Mr. Trump’s campaign is confronting enduring vulnerabilities that make his nomination a considerable risk for his party. Those weaknesses were laid bare in New Hampshire on Tuesday, where independents, college-educated voters and Republicans unwilling to dismiss his legal jeopardy voted in large numbers for his rival, Nikki Haley.

Mr. Trump still won easily. The voters opposed to his bid didn’t outnumber the many Republicans clamoring to see him return to power. But the results, delivered by more than 310,000 voters in a politically divided state, pointed to the trouble ahead for Mr. Trump as the presidential race leaves MAGA world and enters a broader electorate, one that rejected him less than four years ago.

“When I have people come up to me who voted for Reagan in ’76 and have been conservative their whole life say that they don’t want to vote for Trump again, that’s a problem,” Gov. Ron DeSantis of Florida said Tuesday in an interview with Blaze TV, a conservative media company, just a couple of days after he ended his own campaign and endorsed Mr. Trump. “So he’s got to figure out a way to solve that.”