JACKSONVILLE, Fla. — When federal officials inspected this city’s Head Start program five years ago, they found moldy classrooms, exposed wires, leaking sewage, a sagging roof and trash-strewn playgrounds littered with safety hazards. A teacher had jerked a student so hard she dislocated the girl’s shoulder.

The visitors were so alarmed at the neglect that they began changing diapers themselves. What they did next was even more unusual: They fired the nonprofit running the program, the Urban League, and chose a new one.

Now run by Lutheran Services Florida, Jacksonville’s Head Start program has cleaner classrooms, more teachers with college degrees, a full-time teaching coach and rising scores on the federal government’s main yardstick of classroom quality. Once in the lowest 10 percent nationwide, Jacksonville now has scores that approach the national average.