When he started his presidential campaign in April, Pete Buttigieg sounded like he fit in with the most liberal candidates in the Democratic field. He had a plan to overhaul the Supreme Court, talked up abolishing the Electoral College and spoke favorably about a single-payer health-care system.

The 37-year-old mayor of South Bend, Ind., has spent the last several months, however, dispelling any notion he is in the same camp as the two leading progressives in the race, Sens. Elizabeth Warren of Massachusetts and Bernie Sanders...