The UN has urged Afghanistan's neighbours to keep their borders open as the number of civilians fleeing the Taliban onslaught swells.
Thousands of those internally displaced have been arriving in Kabul, seeing the capital as their last safe refuge.
Food shortages are "dire", the World Food Programme (WFP) said. It warned of a humanitarian catastrophe.
On Friday, the Taliban seized the country's second-largest city Kandahar, the latest provincial capital to fall.
The southern city of 600,000 people was once the Taliban's stronghold, and is strategically important because of its international airport, agricultural and industrial output.
The insurgents also took the nearby city of Lashkar Gah, and now control about a third of Afghanistan's provincial capitals.
- LIVE UPDATES: More cities fall as Taliban advance
- EXPLAINER: Kandahar, Afghanistan's turbulent province
- ANALYSIS: An untidy end - or ominous beginning?
- VOICES: Fearing the Taliban, Kabul's young women plead for help
The advance comes as US and other foreign troops withdraw after 20 years of military operations. More than 1,000 civilians were killed in Afghanistan in the past month alone, according to the UN.
Sahraa Karimi, an Afghan filmmaker in Kabul, told the BBC it felt like the world had turned its back on Afghanistan and she feared a return to "dark times".
Life under the Taliban in the 1990s saw women forced to wear the all-covering burka, education restricted for girls over 10 and brutal punishments brought in, including public executions.
"I am in danger - [but] I don't anymore think about myself," Ms Karimi said. "I think about our country... I think about our generation: that we did a lot to bring these changes.
"I think about young girls... there are thousands of beautiful, young talented women in this country."
Many of those seeking safety in Kabul have been sleeping on the streets. About 72,000 children are among those fleeing to the capital in recent days, according to Save the Children.
"We have no money to buy bread, or get some medicine for my child," Asadullah, a 35-year-old street vendor who fled northern Kunduz province after the Taliban set fire to his home, told the BBC.
"All of our home and belongings caught fire, so we came to Kabul and pray to God to help us," added Asadullah, who is now with his family in Kabul.
Makeshift camps have been established on scrubland on the outskirts of the capital, while many others have reportedly been sleeping in abandoned warehouses.
Speaking to the BBC shortly before Kandahar fell, Pashtana Durrani, executive director of an education NGO that works with Afghan girls, said she was scared for her life because of her vocal role in advocating for women's education.
"The girls who we work with have already fled," she said. "I don't know where the students are and I'm personally scared about their life. What if they're married to a Taliban fighter? What will their life be like?"
Afghan war - the basics
- US-led forces toppled the Taliban: In 2001 US-led forces overthrew Afghanistan's Taliban rulers after the 9/11 attacks masterminded by al-Qaeda leader Osama bin Laden, who was based there.
- Twenty years of occupation and military operations followed: The US and allies oversaw elections and built up Afghan security forces, but the Taliban continued to launch attacks.
- Eventually the US made a deal with the Taliban: They would pull out if the militants agreed not to host terrorist groups. But talks between the Taliban and the Afghan government failed. US-led forces withdrew this year and the Taliban have now retaken most of the country.
'From here where else do we run?'
By Yogita Limaye, BBC News, Kabul
People are in disbelief about what's happened in a single day. Five provincial capitals - among them major cities - fell to the Taliban on Thursday.
In Kabul, thousands of people have been arriving - but this is a number that changes by the hours. They've left with very few belongings. These are people, who had homes and jobs, and shops and farms - and they just had to leave everything behind and try to run to safety.
Some of them have taken days, and these are dangerous journeys - past Taliban checkpoints and active frontlines - to get to Kabul. This is the last place many of them believe they can go to. They say, from here where else do we run?
They are angry at the government about being left to fend for themselves. The government says it is going to house them in mosques and provide them with the relief - but there is not enough for everyone who is coming in.
There's anger too that the US and UK are evacuating their own citizens and leaving Afghans to their fate.
The first US troops have begun landing at Kabul airport, part of a 3,000-strong force being sent to help evacuate the country's diplomatic staff, a US defence official told CBS.
The embassy has said it is hearing reports that the Taliban are executing Afghan troops who were surrendering, saying it "could constitute war crimes".
The UK is deploying 600 troops to support British nationals leaving the country. Staff at the British embassy have been reduced to a core team, while Norway and Denmark announced the temporary closure of their embassies.
In a blow to attempts to build a coalition against the militants, the Taliban say they have detained the veteran militia leader Ismail Khan after they seized Herat.
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