About two dozen people aboard the bus were transported for medical treatment, Hart County schools Superintendent Ricky Line said. One person suffered a head injury, Line said, and others had broken bones.
None of the injuries appeared to be life-threatening, Line said.
In addition to the driver, the bus was carrying 34 students and 20 parents. Line said the students, from the Cub Run community, were mostly sixth- to eighth-graders; some were from high school. The trip was not school-related.
"The school didn't have anything to do with it except it's our precious cargo," he said.
The bus was operated by New Image Travel of Evansville, Ind., for WorldStrides.
Rick Juelfs of New Image said company officials were on their way to the scene Wednesday night.
A phone message and email seeking comment from WorldStride were not immediately answered.
Kentucky State Police Trooper Jonathan Biven of the Bowling Green post said the bus crashed on Kentucky Route 728 in Hart County, about 70 miles south of Louisville.
The injured were being transported by ambulance or private vehicle to area hospitals to be checked out.
Hardin Memorial Hospital house manager Jerry Taylor said about seven people were brought there by ambulance with injuries that were not life-threatening.
The bus had traveled just a few miles before the wreck happened.
"It happened in the first seven minutes of a 14- to 15-hour bus trip, and it hadn't even left the county," Line said.
Line said he was relieved the injuries were not more serious and recognized the help of emergency workers, who scrambled ambulances to several different hospitals.