The vehicle – reports say driven by a 16-year-old – lost control and broke through a railing before falling 30 metres (100 ft) into the Banas river.
It was carrying passengers in Rajasthan to the Ramdevji Hindu temple in Sawai Madhopur district, local media report.
Most of the bodies have been retrieved and the injured taken to local hospitals.
Police said the driver was among those who died, describing him as an untrained minor.
Most of the victims, women and children among them, drowned, the authorities say.