The violence erupted in the morning and spread as a new political party formed by students who led the uprising against Hasina in August last year announced a march toward southwestern Gopalganj district, Hasina's ancestral home, and her Awami League party's stronghold.
Authorities later imposed an overnight curfew in the district.
Earlier TV footage showed pro-Hasina activists armed with sticks attacking police and setting vehicles on fire as a convoy of about 20 vehicles carrying the leaders of the National Citizen Party arrived as part of commemorations of the uprising.
Party leaders took shelter in the office of the local police chief as footage showed some of them were being escorted by soldiers to an armoured vehicle for safety.
They later left for a neighbouring district with security escorts.
Jibitesh Biswas, a senior official of a state-run hospital, told reporters that the bodies of at least three people had been brought in.
The English-language Daily Star newspaper reported that four people died.
The interim government said on Wednesday said the attackers would not go "unpunished".
The use of violence in Gopalganj is "utterly indefensible," a statement issued on behalf of interim leader Muhammad Yunus said.
Hasina's Awami League party issued several statements on X condemning the violence.
It blamed the government for killing and injuring the "civilians".
"We urge the world to take note of this blatant use of security apparatus ... and violent use of mob to exterminate dissenters for their ideals and political belief," one Awami statement said.
Nahid Islam, convener of the National Citizen Party, issued a 24-hour ultimatum to the interim government to arrest those responsible for the Gopalganj attacks.
He raised the prospect of another march in the neighbouring district of Faridpur on Thursday.
The conservative Jamaat-e-Islami party condemned the attacks on the student-led party leaders and announced a country-wide protest in all districts and metropolitan cities for Thursday.
Bangladesh has been in political turmoil since Hasina was toppled and fled to India.
Nobel Peace Prize laureate Yunus took over three days after her ouster and pledged to restore order.
He said a new election would be held in April next year.
Gopalganj is a politically sensitive district because Hasina father's mausoleum is located there.
Sheikh Mujibur Rahman, the country's independence leader, was buried there after he was assassinated along with most of his family members in a military coup in 1975.
The National Citizen Party launched its "July March to Rebuild the Nation" on July 1 across all districts in Bangladesh as part of its drive to position itself as a new force in Bangladeshi politics, which as been largely dominated by two dynastic parties - Hasina's Awami League and the Bangladesh Nationalist Party, or BNP, headed by Hasina's rival and former prime minister Khaleda Zia.
