It is estimated that nearly 1,000 people have died in India since the onset of monsoon rains in June.
Several rivers have burst their banks in the north of Bihar, while in other parts, dams have been forced to discharge brimming water.
Satish Chandra Jha, an official from Bihar's relief and rehabilitation department, said more than 3,000 villages in the state were waterlogged.
At least 25 people have died in Bihar in the past week and the figure is expected to rise.
Angered by the apparently slow response of authorities, flood victims in the northern Muzaffarpur district looted a truck loaded with bread on Thursday.
In Assam, where nearly three million out of its 27 million residents are homeless, people clashed with police in several places, demanding food, shelter and medicines.
Many people were injured in the clashes, and police had to fire in the air to stop the violence, officials said.
A 10-year-old boy was shot by police in one protest on Thursday in the west of the state.
Officials were grappling to provide relief in the region where more than 7,000 square km of farmland has been submerged.
Tarun Gogoi, Assam's chief minister, said: "Relief and rescue workers are working on a war footing. But in this kind of a devastating flood, we know we cannot satisfy each and everybody."
He said baby food and water purifying tablets were being dispatched, but people stranded in Senimari village, just 75km east of Assam's main city, Guwahati, had yet to be reached.
Six more people drowned in the state overnight, taking the toll from flood-related deaths in Assam in the past week to 33.
In the Indian capital, New Delhi, a night of heavy rain left many parts of the city under several feet of water, causing traffic chaos.
In neighbouring Bangladesh, thousands of families have moved into about 1,000 relief camps but many others are living on embankments and highways, with cases of diarrhoea and skin infections rising sharply.
Witnesses have reported shortages of baby food, other edibles and drinking water in flood shelters in government buildings and schools, though the government said it had sufficient stocks.
Officials have forecast more flooding, saying that parts of the capital Dhaka could come under water in a few days.