President Gotabaya Rajapaksa has vowed to restore order, in his first national speech since protests began last month.
Ignoring calls to resign, he offered to cede some powers to parliament and name a prime minister, but set no timetable.
His brother quit as PM on Monday amid fury over soaring prices and shortages.
Sri Lankans are desperate as basic items like food and fuel run out or become unaffordable.
"We have come to the protest site despite the curfew," one protester, Chandrasekaran, told BBC Tamil in Colombo. "We are suffering even now. There is no kerosene, no petrol, no diesel, and no power."
Despite a nationwide curfew, there have been two consecutive nights of arson attacks by mobs - many have targeted property belonging to the Rajapaksas and other politicians who are blamed for the mess the country is in.
Shops near Colombo were torched, as well as a resort owned by the son of former Prime Minister Mahinda Rajapaksa.
Rajapaksa, the president's elder brother and a two-time former president, is holed up in a naval base in the north-east for his own safety, the military has confirmed.
At least nine people have been killed and about 200 injured in unrest since Monday. It began when government supporters attacked protesters who are demanding that Gotabaya Rajapaksa leave office.
Opposition politicians have warned the violence could have been staged to give the army a pretext to take power. Rumours of a possible coup have been fuelled by the presence of large numbers of troops with armoured vehicles on the streets.
But the military have denied any such move is planned.
"When there is a dangerous situation in the country, powers are given to the military to deal with it," Defence Secretary Kamal Gunaratne told a news conference.
"Don't ever think that we are trying to capture power. The military has no such intentions."
Sri Lanka had already seen weeks of protests over its dire financial situation, which has caused the Sri Lankan rupee to plunge, provoking severe shortages of basic items such as food, fuel and medical supplies.
The worst trouble overnight was in the north of the capital, Colombo, where rival groups set fire to shops in the town of Negombo.
On Monday night, mobs burned more than 50 houses belonging to politicians, while a controversial museum dedicated to the Rajapaksa family was also razed to the ground in their traditional heartland, Hambantota, in the country's south.
Shops, businesses and offices were shut for a third day on Wednesday under a nationwide curfew in place until Thursday morning.
"Only if Gota [the president] goes will something positive happen," Kannan, another protester, told the BBC Tamil service at Galle Face Green, the main protest site in Colombo.
"I think young people should rule the country," he said.
Dharshini, a fellow protester at the site, agreed.
"Now, one Rajapaksa has resigned. We don't know whether the other one will also go, or whether we will have to live like people in Somalia. Running a household is very difficult," she said.
The streets of the capital, Colombo, bear the evidence of this week's rioting - a heavy police presence, troops on the streets, and overturned and burnt buses after Monday's violence.
Nonetheless, protesters continue to gather despite the curfew at Galle Face Green. They insist President Rajapaksa has grossly mismanaged the economy and must stand aside.
He is the last Rajapaksa family member in office and the resignation of his brother as prime minister did nothing to placate demonstrators or bring calm.
President Rajapaska says he is holding talks with other political parties aimed at forming a unity government. But the main opposition has said it will not be part of the interim administration unless the president stands down.
At present, there is no clarity on which political parties might come together to form such a government.
The political stalemate comes as Sri Lanka attempts to iron out a bailout package with the International Monetary Fund (IMF) - the island's $81b economy is near bankruptcy.
It has suspended its foreign debt payments, largely because it cannot service loans from China that paid for massive infrastructure projects.
While the pandemic hit the vital tourism sector and shrank Sri Lanka's earnings and foreign exchange reserves, experts say problems have been exacerbated by populist tax cuts in 2019 and a disastrous ban on chemical fertilisers in 2021 that devastated crop yields.
Protesters gathered in front of Trincomalee Naval Base in the north-east after reports that Mahinda Rajapaksa had fled there with his family after escaping from his Colombo residence when it was besieged by crowds on Monday night.
The military confirmed on Wednesday that he was inside the base.
"We took Mahinda to the naval base for safety reasons," the defence secretary said.
Earlier, there had been rumours that Rajapaksa and others in the family had fled to India, which the Indian High Commission in Colombo denied.