The driver lost control as the bus, with 28 people on board, was crossing a bridge, according to Bundi district police superintendent Shrivaj Meena.
"The bus was carrying a marriage party from Kota. The bus fell off a bridge," Meena told AFP.
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Villagers and rescue workers were seen in images broadcast from the scene trying to retrieve bodies from the river as the bus lay upside down in the water.
India records more than 150,000 road deaths each year which are widely blamed on badly maintained vehicles and reckless driving.
Last month, 26 people were killed when a bus collided with an auto rickshaw and fell into a well in western India.
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