WASHINGTON—After weeks in the news because of her disputes with Speaker Nancy Pelosi and being on the receiving end of insults from President Trump, Rep. Ilhan Omar marched out of the Capitol Thursday afternoon, bringing traffic to a halt on Independence Avenue by the Mall, and told a gaggle of reporters not to stampede a photographer who fell in the melee.

“For the sake of a story, don’t kill each other,” she said.

As...