The incident took place early morning in Soorwal police station area when the bus carrying about 45 passengers was on its way to Lalsot from Sawai Madhopur, they said.
The driver of the bus lost control over the vehicle and it plunged into the river after breaking the railing of the bridge, the police said.
Prime Minister Narendra Modi expressed grief over the incident and said his thoughts were with the families of the deceased.
Soorwal police station SHO Anoop Singh said of the 33 people killed, seven are women and four children. The injured were rushed to hospitals.
The bus driver was also killed in the incident, a police officer had said earlier.
He said that 19 bodies were handed over to the kin of victims after post-mortem and autopsy of 14 bodies was yet to be conducted.
One body is still unidentified, he added.
On reports that the bus driver was a minor, the SHO said it was yet to be ascertained.
Rajasthan Chief Minister Vasundhara Raje expressed her condolences to the families of the victims.
"I am saddened by the bus accident that took place in Dubi Banas. Took stock of the situation from the officials present on the spot and directed them to provide all possible assistance to the affected people," she said in a tweet.
In a statement issues later, she said in-charge minister of the district Rajpal Singh Shekhawat was rushed to the spot to oversee the relief measures.
Union minister Rajyavardhan Singh Rathore, Rajasthan Congress Chief Sachin Pilot, Sawai Madhopur MLA Diya Kumari and NPP leader Kirori Meena also expressed grief over the incident.
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