NATO has called on the Taliban to allow people being evacuated to leave Afghanistan, and vowed that the allies would remain in "close coordination" while operations continue.
The joint declaration by NATO's 30 member countries was made following an emergency videolink conference of their foreign ministers to discuss evacuation efforts and the next steps to take.
"We call on those in positions of authority in Afghanistan to respect and facilitate their safe and orderly departure, including through Hamid Karzai International Airport in Kabul," they said.
"As long as evacuation operations continue, we will maintain our close operational coordination through allied military means at" the airport.
The United States and NATO allies have been flying their citizens, Afghan staff and their families out of Kabul since the weekend, when the Taliban overran the capital and regained power.
While thousands have been evacuated so far, NATO chief Jens Stoltenberg said at the start of the videoconference that foreigners and Afghans were struggling to reach Kabul airport.
Earlier, a NATO official said that more than 18,000 people have been flown out of Kabul.
The US State Department is expected to announce that its evacuation flights out of Kabul will now be able to land in Europe. US officials told Reuters that this was because of an overflow of people in Qatar.
The US is aiming to ramp up the effort to get thousands of US citizens and at-risk Afghans out of Taliban rule. The State Department declined to comment but said the agency is "grateful to all of our partners who are playing a role in this effort".
The Western alliance halted its own mission in Afghanistan in May after the United States decided to withdraw its troops, but still has up to 800 civilian staff helping run Kabul airport.
Several countries have mounted missions to airlift their nationals and local allies to safety, but Mr Stoltenberg warned the allies must work together and the Taliban must cooperate.
One of the main challenges with evacuating Afghanistan is getting people into the airport and processed, says NATO Secretary General Jens Stoltenberg, not a lack of planes | Read: https://t.co/2xASHDJ0rX pic.twitter.com/kRLXa3gIp0
— RTÉ News (@rtenews) August 20, 2021
"Hundreds of NATO staff and contractors have also been playing a vital role," he said, thanking US, UK and Turkish troops for protecting NATO's staff in Kabul and at the airport.
"They have kept Kabul airport operational, including air traffic control, essential airport services and communications," he said.
"The main challenge we face however, is ensuring that people can reach and enter Kabul airport," he told the ministers, after reports that the Taliban are hunting Afghans who worked with NATO.
"We expect the Taliban to allow for the safe passage of all foreign nationals and Afghans seeking to depart the country. This is the most urgent task today."
It comes amid growing concern for people in Afghanistan who helped the international coalition after an intelligence paper for the UN reported that the Taliban are hunting them down, house by house.
It is believed that the Taliban is using social media to track down police officers, soldiers, intelligence officers and their families.
To combat this Facebook has said it has introduced safety measures that allows users in Afghanistan to lock their account.
After routing government forces and taking over Kabul on Sunday, the hardline Islamist movement's leaders have repeatedly vowed a complete amnesty as part of a well-crafted PR blitz.
Women have also been assured their rights will be respected, and that the Taliban will be "positively different" from their brutal 1996-2001 rule.
But with thousands of people still trying to flee the capital aboard evacuation flights, the report for the United Nations confirmed the fears of many.
The Taliban have been conducting "targeted door-to-door visits" of people who worked with US and NATO forces, according to a confidential document by the UN's threat assessment consultants seen by AFP.
The report, written by the Norwegian Center for Global Analyses, said militants were also screening people on the way to Kabul airport.
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"They are targeting the families of those who refuse to give themselves up, and prosecuting and punishing their families 'according to Sharia law'," Christian Nellemann, the group's executive director, said.
"We expect both individuals previously working with NATO/US forces and their allies, alongside with their family members to be exposed to torture and executions."
The Taliban have denied such accusations in the past and have several times issued statements saying fighters were barred from entering private homes.
They also insist women and journalists have nothing to fear under their new rule, although several media workers have reported being beaten with sticks or whips when trying to record some of the chaos seen in Kabul in recent days.
During their first stint in power, women were excluded from public life and girls banned from school. People were stoned to death for adultery, while music and television were also banned.
The United States invaded Afghanistan and toppled the group in 2001 following the 11 September attacks for providing sanctuary to al-Qaeda.
A video posted online by a high-profile woman journalist this week for a government-run television station offered a different reality to the Taliban's new image of tolerance.
"Our lives are under threat," Shabnam Dawran, an anchor in state-owned broadcaster RTA, said as she recounted being barred from the office.
"The male employees, those with office cards were allowed to enter the office but I was told that I couldn't continue my duty because the system has been changed," she said.
There have been isolated signs of opposition to the Taliban in parts of Afghanistan this week.
Small groups of Afghans waved the country's black, red and green flags in Kabul and a handful of suburbs yesterday to celebrate the anniversary of Afghanistan's independence - on occasion in plain sight of patrolling Taliban fighters.
Taliban fighters fired guns to disperse dozens of Afghans in Jalalabad who waved the flag on Wednesday.
Russia also emphasised yesterday that a resistance movement was forming in the Panjshir Valley, led by deposed vice-president Amrullah Saleh and Ahmad Massoud, the son of a killed anti-Taliban fighter.
"The Taliban doesn't control the whole territory of Afghanistan," Russian Foreign Minister Sergei Lavrov said.
In the Panjshir Valley northeast of Kabul, Ahmad Massoud, said he was "ready to follow in his father's footsteps".
"But we need more weapons, more ammunition and more supplies," he wrote in the Washington Post.
Taliban kill relative of DW journalist in Afghanistan: broadcaster
Taliban fighters in Afghanistan have shot and killed a relative of a Deutsche Welle journalist while hunting for him, the German public broadcaster said.
The militants were conducting a house-to-house search for the journalist, who now works in Germany, DW said yesterday.
A second relative was seriously wounded but others were able to escape, it said, without giving details of the incident.
DW director general Peter Limbourg condemned the killing, which he said showed the danger to media workers and their families in Afghanistan.
"The killing of a close relative of one of our editors by the Taliban yesterday is inconceivably tragic, and testifies to the acute danger in which all our employees and their families in Afghanistan find themselves," he said.
"It is evident that the Taliban are already carrying out organized searches for journalists, both in Kabul and in the provinces. We are running out of time!"
The Taliban had raided the homes of at least three other DW journalists, the broadcaster said.
DW and other German media organisations have called on the German government to take swift action to help their Afghan staff.
