The accident happened when the driver lost control near the mountainous village of Marotthi, around 250 kilometres south of the main city of Srinagar.
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"We have pulled out 23 bodies so far and evacuated around 30 other injured passengers, some of whom are critically injured," Danish Rana, a local inspector general of police, told AFP.
"The bus was apparently overloaded," added Rana, while warning the death toll could rise further.
Read: Fatalities: Two men killed in road mishap
Some of the injured passengers were airlifted by helicopter to hospital.
India has some of the world's deadliest roads, with more than 200,000 fatalities annually, according to the World Health Organization.
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