Fighting is raging around three major cities in south and west Afghanistan, as Taliban militants seek to seize them from government forces.
Herat, Lashkar Gah and Kandahar saw continued clashes on Sunday.
The Taliban have made rapid rural gains since it was announced almost all foreign troops would go by September.
But the fate of these key cities could be crucial amid fears of a humanitarian crisis and how long government forces will be able to hold out.
The fundamentalist Islamist militia is already thought to have captured up to half of all Afghanistan's territory, including lucrative border crossings with Iran and Pakistan, but it has yet to take a provincial capital.
In Lashkar Gah, heavy fighting continued inside the city on Sunday.
Insurgents were reportedly only a few hundred metres from the governor's office on Saturday - but had been pushed back by nightfall.
Afghan and US air strikes have reportedly targeted Taliban positions and government forces say they have killed dozens of militants.
One resident, Halim Karimi, told Agence France-Presse: "Neither the Taliban will have mercy on us, nor will the government stop bombing."
Analysis by Secunder Kermani, BBC News
The Taliban's firm focus is now on Afghanistan's cities. The situation is fluid but Lashkar Gah, capital of Helmand province where many US and British soldiers lost their lives, looks the most vulnerable right now. Pro-Taliban social media accounts have uploaded videos of their fighters in the heart of the city.
Afghan special forces are being sent in to help push them back, but one local resident told us even if that does end up happening, the Taliban's advance is a powerful assertion of their strength.
The militants are understood to have taken up some positions in the homes of ordinary families, which will make it harder to dislodge them. More lengthy and bloody fighting looks to be ahead.
Flights at Kandahar airport were suspended on Sunday after Taliban rockets struck the complex before dawn, causing some damage to the runway.
One Kandahar MP told the BBC on Saturday the city was at serious risk of falling to the Taliban, with tens of thousands of people already displaced and a humanitarian disaster looming.
Gul Ahmad Kamin said the situation was getting worse hour by hour, and the fighting within the city was the most severe in 20 years.
He said the Taliban now saw Kandahar as a major focal point, a city they want to make their temporary capital. If it fell, then five or six other provinces in the region would also be lost, Mr Kamin said.
He said the Taliban fighters were on several sides of the city and because of the large civilian population government forces would not be able to use heavy weaponry if the militants got fully inside.
Afghan special forces have been deployed in the economically important city of Herat, and the situation appeared more stable on Sunday.
Afghan troops are fighting alongside the veteran warlord and anti-Taliban commander, Ismail Khan, who has mobilised citizens to take on the militants.
Air strikes have also been carried out on Taliban positions outside the city.
A guard outside a UN compound near the airport was killed on Friday in what the UN described as a deliberate Taliban attack.
The EU's special envoy for Afghanistan, Tomas Niklasson, said he believed the war was set to get much worse.
He said he feared the Taliban way of thinking now was "something they had in the past - re-establishing... their Islamic emirate".
And the former head of the British Armed Forces, Gen David Richards, warned the international withdrawal could result in the collapse of the Afghan army's morale, leading to Taliban control and possibly a renewed international terrorist threat.
Humanitarian organisations have also warned of a major crisis in coming months as the Taliban continue their offensive - with a lack of food, water and services, and overcrowding in camps for the displaced.
US troops and their Nato and regional allies forced the Taliban from power in November 2001.
The group had been harbouring Osama Bin Laden and other al-Qaeda figures linked to the 11 September 2001 attacks in the US.
But despite a continued international presence, billions of dollars of support and training for the Afghan government forces, the Taliban regrouped and gradually regained strength.
In February 2020, then-US President Donald Trump and allies agreed to formulate a deal with the Taliban on the withdrawal of international combat forces.
This year, President Joe Biden announced the withdrawal would take place by September.