The survivors were initially taken to Valley View Medical Center in Cedar City, hospital spokesman Scott Monroe said. Seven of them, including one child, were flown to Salt Lake City area trauma centers for additional care, he said.
Three people remained in fair condition at Valley View on Tuesday and two had been treated for minor injuries and released, Monroe said.
He said doctors were using a telephone interpreter service to communicate with the injured passengers.
Of the passengers flown to Salt Lake City, two men and two women were in critical condition at Intermountain Medical Center, hospital spokesman Jess Gomez said.
Two other women in critical condition were at University Hospital.
The child was taken to Primary Children's Medical Center but officials there said they could not release any information.
The bus was heading to Bryce Canyon National Park when it veered into a median and rolled over at around 6:40 p.m. Monday about four miles north of Cedar City, Utah Highway Patrol spokesman Cameron Roden said.
The van-sized bus ended up in a mangled heap on its top at the crash site about 250 miles southwest of Salt Lake City.
Trooper James Keller told The Spectrum of St. George at least two passengers were ejected from the bus, and one passenger trapped inside was killed.