Police said the accident took place between 6.30 am and 7 am, when the bus was overtaking another vehicle on a bridge on the Sawai Madhopur-Lalsot road.
The driver of the bus lost control over the vehicle and it fell into the river after breaking the railing of the bridge, the police said.
The driver's body is reportedly among the 33 so far been accounted for.More news: Arsenal 3 Liverpool 3: Points shared in classic Christmas cracker
'Sad to learn about the accident at Sawai Madhopur.
According to local media reports, the bus was carrying 50 pilgrims from central Madhya Pradesh and northern Uttar Pradesh states to the Ramdevra Temple in Rajasthan.
Chief minister Vasundhara Raje expressed grief over the deadly incident and directed the authorities to gear up the rescue operation. Those injured were rushed to the government hospital, while the bodies of the deceased were being pulled out of water.