“I’d be lying to you if I told you it was fun,” he said.

In California, Representative Tom McClintock was escorted by police officers after a town-hall-style meeting earlier this month; in Utah, the crowd chanted “Do your job!” at Representative Jason Chaffetz, the chairman of the Oversight Committee. At a meeting last week, House Republicans were advised on security precautions so they would be prepared for protesters at town-hall-style meetings or their district offices.

The questions from voters on display this weekend at a series of town-hall-style meetings in Wisconsin’s Fifth Congressional District, many of which were focused on the future of the health care law, underscored the quandary many lawmakers are facing even in solidly Republican districts.

The imminent problem: Constituents want answers, and without any consensus on how to go about replacing the law, Republicans have little to say.

“It’s kind of like, you know, getting a 30,000-piece jigsaw puzzle for Christmas,” Mr. Sensenbrenner said, “and, you know, cleaning off the dining room table and seeing how long it takes to put all the 30,000 pieces together in the right place. It’s not going to be easy.”

Mr. Sensenbrenner won re-election last year by 37 percentage points. His right-leaning district, which includes suburbs around Milwaukee, voted decisively for Mr. Trump over Hillary Clinton.

At three town-hall-style meetings over the weekend, Mr. Sensenbrenner sat at the front of the room to take questions from people who submitted slips of paper listing their name and address. When he called on people, he read their names and where they live — a practice that makes people “less likely to make fools of themselves,” he said in the interview.