Pete Buttigieg is rising in the Democratic presidential polls on his rhetorical gifts, distinctive resumé and fund-raising prowess. But the 37-year-old mayor of South Bend, Indiana, has so far largely received a pass on his government record from the press and even other Democrats. That isn’t likely to continue, and a good place to start is his trouble controlling violent crime.

Violence has dropped sharply throughout the U.S. over the past two decades as a result of improved policing, more incarceration and demographic...