Several vehicles and a number of houses, including an office of the main opposition Awami League, were also ransacked in the northern town of Netrokona, 200km from Dhaka, on Wednesday.
The violence erupted after police opened fire on thousands of people marching towards the office of the district administrator to protest over the sharp rise in prices of food, fuel and other essentials in recent weeks.
Ashraf Ali Khan, a local Awami League leader, said: "The violence flared up when stick wielding pro-government activists attacked the protesters."
At least 20 people have been killed during protests in the country's northwest in recent months over demands for power, water and diesel to run irrigation pumps on farms.
Over the past five days, nearly 130 people have been injured during protests over power and water in the capital, Dhaka.
Nearly half of Bangladesh's power stations are not working because of mechanical faults.
In Dhaka, a city of 10 million people, the Water and Sewerage Authority can supply only two-thirds of demand every day.