In a third besieged city, Herat, in the west, government commandos are battling the insurgents after days of fierce fighting. Government forces have taken back some areas after a UN compound was attacked on Friday.
As government forces struggled to contain Taliban advances, President Ashraf Ghani blamed the sudden withdrawal of US troops for the increase in fighting.
“The reason for our current situation is that the decision was taken abruptly,” Ghani told parliament.
Mr Ghani said he had warned Washington that the withdrawal would have “consequences”.
Although nearly all its military forces have left, the US has continued its air offensive in support of government troops.
President Biden’s administration announced on Monday that because of the increase in violence, it would take in thousands more Afghan refugees who worked with US forces.
The US and UK have accused the Taliban of “massacring civilians” in a town they recently captured near the Pakistan border.
Gruesome videos that emerged from Spin Boldak apparently showed revenge killings. The Taliban have rejected the accusations.
“The Taliban’s leadership must be held responsible for the crimes of their fighters,” the US embassy in Kabul tweeted.
“If you cannot control your fighters now, you have no business in governance later.”