The accident took place at Basaitha chowk under Bennipatti police station in Madhubani, about 50 km from the district headquarters.
Deputy Superintendent of Police (DSP) Benipatti Nirmala Kumari said four bodies have been fished out so far and search is on for others.
About 65 passengers were on board the bus. Some of them swam to safety while over a dozen are feared to have been killed.
The private bus was on way to Sitamarhi from Madhubani when the mishap took place on state highway.
The SP and other policemen are stranded at the accident site and waiting for police reinforcement to reach the spot.
Meanwhile, in Patna, Chief Minister Nitish Kumar expressed deep sorrow over death of passengers in the bus mishap at Madhubani.
The CM said in a statement that reason of the accident was being looked into. He instructed officials to speed up relief and rescue operation.
Kumar instructed Revenue minister Madan Mohan Jha and Panchayati Raj minister Kapildeo Kamat to rush to the accident spot.
The CM issued instruction to provide all possible assistance to victims of the bus tragedy.
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