Another three people were in critical condition after the bus, carrying 35 people, veered off the bridge in the mountainous northern state of Uttarakhand.
"The bus collided with the bridge and overturned" into a small river, police officer Mukhtaar Hasan told AFP. Some of the victims appeared to have drowned, police said.
The accident occurred in Juyalgarh village, some 150 kilometres from state capital Dehradun.
India has the highest number of road accidents globally, with 138,000 recorded in 2012, the latest year for which official government figures are available.
Transport analysts attribute the huge number of accidents to poor roads, ill-trained drivers and reckless driving.
The national government has recently put forward proposals for new legislation to make India's road safer by stiffening lax traffic regulations.
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