Chaotic scenes have been reported outside Kabul international airport as governments rush to bring home their citizens as well as Afghan colleagues.
Taliban militants controlling access to the airport have fired shots into the air to disperse approaching crowds.
Some 5,000 people have been evacuated in the last 24 hours through the airport, which is being run by specially deployed US troops.
It comes three days after the Taliban takeover of Afghanistan.
"It's absolutely hectic and chaos out there," one Western official told the Reuters news agency, as thousands of people remain desperate to leave, including Afghans who helped the US-led mission in the country over the past 20 years.
Taliban fighters deployed around the airport are reportedly not permitting Afghans without documents to enter. They are, however, allowing entry to people with American passports, according to US officials.
The hardline Islamist group's dramatic takeover of Kabul on Sunday took many Western governments by surprise.
Staff at the Dutch embassy have faced criticism after saying they did not have time tell Afghan colleagues they were going.
The former president of the Dutch military union feared there was little time left to evacuate interpreters and local staff: "If we don't succeed in the next 48 to 72 hours it'll be too late," Anne-Marie Snels told the BBC.
Dozens of planes from around 15 countries have reportedly been sent to Kabul to assist with the evacuations. American, French, Dutch, German, Spanish and British aircraft have all left in recent hours.
The US military, which is planning to evacuate more than 30,000 people by the end of the month, is guarding the runways and has taken over air traffic control.
Its troops fired shots overnight as part of crowd-control efforts but no-one was injured, Pentagon spokesman John Kirby said on Wednesday.
He said 4,500 US troops were currently at the airport, but hundreds more were set to arrive in the coming hours.
"We have an obligation to get as many people out of Afghanistan as we can," Mr Kirby said, adding that the US was aiming to increase its evacuation capacity to more than 5,000 people a day.
About 11,000 US citizens remain in the country, AFP news agency reports. Two thousand people have been evacuated by the US in the past 24 hours, including 325 Americans.
France, Germany, Spain, Poland and the Czech Republic all said they had successfully evacuated some of their citizens as well as Afghans.
Some 700 British nationals and Afghans were airlifted out on Tuesday and the UK was "trying to scale up the speed and pace over the next couple of days", UK Ambassador Laurie Bristow said.
Evacuee's fear and relief
One British aid worker told the BBC that she felt lucky to escape. "The runways were crowded with hundreds of Afghan families hoping to get out somehow," Kitty Chevallier told the PA news agency,
She added that she was aware of how "immensely lucky" she was to have been helped out, when others, including friends and colleagues, remained stranded.
Elsewhere, one woman who left Afghanistan on a German evacuation plane spoke of her fear before boarding.
"We had to force our way through and my little son fell over and we were scared but we made it," she told Reuters. "An American guy showed goodwill and realised we were totally exhausted."
"He said 'all good, you may go in'. Others behind cried and lay on the ground. It was scary," she said.
The unfolding crisis has prompted wider calls for Western countries to resettle Afghan refugees.
"We cannot abandon [Afghan colleagues] and we are doing everything we can to offer them shelter," the European Union's foreign policy chief, Josep Borrell, said on Wednesday.
German Interior Minister Horst Seehofer, however, was among a number of European leaders to warn of an influx of refugees from Afghanistan, suggesting between 300,000 and five million people could head to Europe.
The UK has said it will settle 20,000 Afghan refugees over the next few years, while Austrian Interior Minister Karl Nehammer said on Wednesday that "the aim must be to keep the bulk of people in the region".
French President Emmanuel Macron has come under fire from opponents for saying Europe should put together a "robust initiative to thwart illegal migration". He accused critics of twisting his comments and said "France does and will continue to do its duty to protect those who are in the most danger".