DES MOINES, Iowa—It’s the sort of problem many candidates would envy. Sen. Bernie Sanders is drawing large, ebullient crowds that are taxing an upstart presidential campaign that wasn’t expected to go very far.

The Vermont independent, a favorite of the Democratic Party’s liberal wing, is being feted by standing room-only audiences that in some cases surpass those of front-runner Hillary Clinton.

More than 3,000 came to a Sanders speech in Minneapolis in May; 700 attended his speech at Drake University here Friday night, about the same number who went to a Hillary Clinton event on Sunday that featured a buffet table and a live band. More than 3,000 people have RSVP’d for a Sanders rally in Denver on Saturday, the campaign says.

A Suffolk University poll released Tuesday showed Mrs. Clinton with a surprisingly narrow lead over Mr. Sanders—41% to 31%—in the early-voting state of New Hampshire. In May, a Bloomberg/St. Anselm College poll had found Mrs. Clinton with a 44-point lead in the state.

The Bernie boomlet is forcing the campaign to improvise. Aides have set up loudspeakers for people left outside Sanders events, and scrambled to find larger venues to accommodate unexpected crowds who relish his attacks on what he calls the “cocky billionaire class.”