A truck was firebombed in the northern city of Bogra, killing the driver and his passenger, as activists enforced month-long transport blockade, police chief Saifuzzaman Chowdhury told AFP.
Officers shot dead a protester in the capital Dhaka after he threw a petrol bomb at a police patrol van, police said.
And a police constable died of injuries suffered last month during another attack blamed on the main opposition Bangladesh Nationalist Party, led by two-time former premier Ms Khaleda Zia.
"He was among several officers who were injured when a petrol bomb was thrown at a police bus on January 17 by protesters," Dhaka Metropolitan Police spokesman Mr Masudur Rahman said.
The deaths take the toll to 62, mostly victims of fire bombings of buses, cars and lorries, as activists try to enforce the nationwide blockade of roads, railways and waterways.
Ms Zia ordered the blockade on January 5 to try to force her arch rival Ms Hasina to call a fresh general election. The BNP and its Islamist allies boycotted the last January 2014 poll as they believed the result would be rigged.
The police constable was the first victim from Bangladesh's security forces since the start of the unrest, which shows no signs of abating.
The government has deployed thousands of security forces across the country and arrested some 10,000 protesters as part of a crackdown on the unrest.
Thousands more demonstrators have gone into hiding and Ms Zia has remained holed up in her office, after her internet, mobile phone and satellite television connections were cut.
Ms Hasina has said elections would not be held before 2019, and on Wednesday she ruled out declaring a state of emergency.
The EU, the nation's biggest export destination, has urged the government and opposition to hold talks to resolve the crisis.
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