NASHUA, N.H.—Democrats in search of a centrist nominee to challenge President Trump are increasingly worried after contests in Iowa and New Hampshire consolidated liberal support behind Bernie Sanders and divided the larger moderate vote among multiple candidates.

The party’s center lane is crowded with the rising former South Bend, Ind., Mayor Pete Buttigieg, Sen. Amy Klobuchar and former Vice President Joe Biden. Looming in the March 3 Super Tuesday states and beyond is former New York Mayor Mike Bloomberg, buoyed by campaign...