The unrest comes as student leaders have declared a campaign of civil disobedience to demand that Prime Minister Sheikh Hasina step down.
Thirteen police officers were also killed when thousands of people attacked a police station in the district of Sirajganj, police said.
The student protest started with a demand to abolish quotas in civil service jobs last month, but has now turned into a wider anti-government movement.
Protesters blocked a highway in Bangladesh's capital city
Both police and some supporters of the governing party were seen shooting at anti-government protesters with live ammunition. Police also used tear gas and rubber bullets.
The total death toll since the protest movement began in July now stands at over 270.
A nationwide overnight curfew has been in place since 18:00 (12:00 GMT).
On Sunday Law and Justice Minister Anisul Huq told the BBC’s Newshour programme that authorities were showing “restraint”.
“If we had not shown restraint there would have been a bloodbath. I guess our patience has limits,” he added.In the capital, Dhaka, access to internet on mobile devices has been suspended.
Deaths and injuries have been reported across the country, including the northern districts of Bogra, Pabna and Rangpur.
“The whole city has turned into a battleground,” a policeman, who asked not to be named, told AFP news agency. He said a crowd of several thousand protesters had set fire to cars and motorcycles outside a hospital.
Asif Mahmud, a leading figure in the nationwide civil disobedience campaign, called on protesters to march on Dhaka on Monday.
"The time has come for the final protest", he said.
Students Against Discrimination, a group behind the anti-government demonstrations, urged people not to pay taxes or any utility bills.
The students have also called for a shutdown of all factories and public transport.
Some of the wounded were driven away by protesters
Around 10,000 people have been reportedly detained in a major crackdown by security forces in the past two weeks. Those arrested included opposition supporters and students.
Some ex-military personnel have expressed support for the student movement, including ex-army chief General Karim Bhuiyan, who told journalists: “we call on the incumbent government to withdraw the armed forces from the street immediately.
He and other ex military personnel condemned "egregious killings, torture, disappearances and mass arrests".
The next few days are seen as crucial for both camps.
The protests pose a momentous challenge to Ms Hasina, who was elected for a fourth consecutive term in January elections, boycotted by the main opposition.
Students took to the streets last month over a quota that reserved one third of civil service jobs for relatives of the veterans of Bangladesh’s independence war with Pakistan in 1971.
Most of the quota has now been scaled back by the government following a government ruling, but students have continued to protest, demanding justice for those killed and injured. Now they want Ms Hasina to step down.
Supporters of Ms Hasina have ruled out her resignation.
Earlier, Ms Hasina offered unconditional dialogue with the student leaders, saying she wanted the violence to end.
“I want to sit with the agitating students of the movement and listen to them. I want no conflict," she said.
But the student protesters have rejected her offer.
Ms Hasina called in the military last month to restore order after several police stations and state buildings were set on fire during the protests.
The Bangladeshi army chief, General Waker-Uz-Zaman, held a meeting with junior officers in Dhaka to assess the security situation.
“Bangladesh Army has always stood by the people and will continue to do so for the interest of people and in any need of the state," Gen Zaman said, according to a release by the Inter Services Public Relation Directorate.
Bangladeshi media says most of those killed in last month’s protests were shot dead by police. Thousands were injured.
The government argues that police opened fire only in self-defence and to protect state properties.
