When in power from 1996-2001, they severely restricted women's rights, staged public executions and blew up ancient Buddhist statues.
But a report by a Norwegian intelligence group said the Taliban had begun rounding up Afghans on a blacklist of people linked to the previous administration or US-led forces that supported it.
Complaints by some Afghan journalists have meanwhile cast doubt on assurances that independent media would be allowed.
It was unclear if the casualties in Asadabad resulted from Taliban firing or from a stampede.
"Hundreds of people came out on the streets," witness Mohammed Salim said. "At first I was scared and didn't want to go but when I saw one of my neighbours joined in, I took out the flag I have at home.
"Several people were killed and injured in the stampede and firing by the Taliban."
Protests flared in the city of Jalalabad and in Paktia province, also in the east.
First Vice President Amrullah Saleh, who said on Tuesday he was the "legitimate caretaker president" after President Ashraf Ghani fled, said on Twitter: "Salute those who carry the national flag and thus stand for dignity of the nation."
In an op-ed for the Washington Post, Ahmad Massoud, the son of guerrilla leader Ahmad Shah Massoud, who was killed by suspected al Qaeda militants in 2001, said he was "ready to follow in my father's footsteps, with mujahideen fighters who are prepared to once again take on the Taliban".
Qatar could host fresh talks between the Taliban and the Afghan government as soon as next week to reach an agreement over power sharing and government transition, said two sources familiar with the process and two foreign diplomats.
G7 foreign ministers called for a united response to prevent the crisis escalating further, in comments echoed by countries including Russia. China said the world should support, not pressure, Afghanistan.
US President Joe Biden said the Taliban must decide if they want international recognition.
"Do they want to be recognised by the international community as being a legitimate government? I'm not sure they do," Biden said in TV interview.
At the border with Pakistan, traders and officials said commercial traffic was starting to return to normal.
Kabul has been largely calm, but 12 people have been killed in and around the airport, a NATO and a Taliban official said.
A Taliban official said they could not be blamed for the chaos at the airport. In one scene captured on social media a small girl was hoisted over the airport's perimeter wall and handed to a US soldier.
The Taliban were "keeping their word" by providing foreign powers with support in evacuations, another Taliban official told Reuters. "We are facilitating safe exit passage not just for foreigners but also to Afghans."
Afghanistan's Ariana News reported that an Afghan national team died in a fall from a US plane at Kabul airport on Monday, when crowds of people were seen trying to board a moving aircraft.
The US military said more than 5,200 US troops were guarding Kabul airport, where multiple gates are now open, while US fighter jets were flying over the city to ensure security for the evacuation operation.
Under a pact negotiated by former President Donald Trump's administration, the United States agreed to withdraw its forces in exchange for a Taliban guarantee not to attack departing foreign forces or let Afghanistan be used for terrorist attacks.
Biden said US forces would remain until all Americans were evacuated, even if that meant staying past an Aug 31 US deadline for withdrawal.