In the capital Dhaka, traffic was thin as police in riot gear used batons to strike several people in a crowd of 1000 anti-government protesters.
Demonstrators later scuffled with activists from the Bangladesh Nationalist Party (BNP), the main partner in the ruling coalition government.
BNP supporters also beat up two journalists covering the demonstration; they were later hospitalised.
In the south-eastern port city of Chittagong more than 2000 paramilitaries and police from armed units were deployed at key facilities such as oil refineries, a spokesman said.
The city was working normally following a strike on Sunday but traffic was light as citizens fearing possible violence opted to stay at home, witnesses said.
The strikes and violent demonstrations which have continued across Bangladesh for a second day came in response to Saturday's attack on a rally of the main opposition Awami League.
In response to the attack, the Awami League called a nationwide general strike for Tuesday and Wednesday while its youth wing said all educational facilities should close indefinitely.
Saturday's attack occurred as Awami League leader Shaikh Hasina Wajid was finishing a speech outside the party's headquarters in Dhaka.
Protestors have been urged to
participate in the strike
The opposition leader escaped with minor leg injuries but her bodyguard was killed, officials said.
"I heard a big bang and the next moment blood splashed on my body," the English language Daily Star quoted her as saying on Monday.
"Leaders and security members formed a human shield around me and helped me into my car. Then came bullets hitting my car one after another," she said, adding that she held the government responsible for the attack.
Party officials have said the blasts were an attempt to assassinate the opposition leader.
The government condemned the attacks on Sunday, strongly denying involvement and appointing a senior High Court judge to carry out a judicial inquiry into the blasts, the official BSS news agency said.
Awami League secretary general Abdul Jalil said the number of dead had risen to 19 following the death Sunday of another blast victim at a private hospital.
Doctors at two Dhaka hospitals said on Monday 21 people were still being treated for serious injuries although an unknown number had also gone to private clinics.
Shaikh Hasina met foreign envoys to Bangladesh, including British High Commissioner Anwar Choudhury, who was injured in a bomb blast in the north-eastern city of Sylhet in May.
US Secretary of State Colin Powell
expressed US support for Hasina
"We were shocked and horrified at this event," Choudhury said after meeting Shaikh Hasina, a former prime minister of Bangladesh.
US Secretary of State Colin Powell telephoned Shaikh Hasina Sunday to express his support, BSS said.
In a statement issued in Washington late Sunday he also condemned the attack and called for calm.
No one has claimed responsibility for the attacks yet.