NEWTOWN SQUARE, Pa.—A series of Democratic wins in the Philadelphia suburbs highlights the challenges President Trump faces in a key general election state.

Those suburban counties backed Democrats in early November local elections, giving them control of the Delaware County Council for the first time since the Civil War, as well as county government in Bucks and Chester counties. While Mr. Trump was not on the ballot this year, the results served as a reminder of Democratic energy in the suburbs, also reflected in Kentucky...