The United States advised Americans in Afghanistan to avoid traveling to Kabul airport on Saturday as thousands try to flee the country, the U.S. Embassy in Afghanistan said in a travel advisory.

"Because of potential security threats outside the gates at the Kabul airport, we are advising U.S. citizens to avoid traveling to the airport and to avoid airport gates at this time unless you receive individual instructions from a U.S. government representative to do so," said the advisory.

The chaos at the airport was not the responsibility of the Taliban, an official from the group told Reuters, adding, "the West could have had a better plan to evacuate." 

Switzerland postponed a charter flight from Kabul because of the chaos, its foreign ministry said. 

"The security situation around Kabul airport has worsened significantly in the last hours. A large number of people in front of the airport and sometimes violent confrontations are hindering access to the airport," the Swiss Federal Department of Foreign Affairs said in a statement. 

The Taliban completed their sudden advance across the country as U.S.-led forces pulled out, coinciding with what German Chancellor Angela Merkel said on Saturday was the "breathtaking collapse" of the Afghan army. 

Images circulated on social media this week of Afghans rushing towards a U.S. C-17 transport plane and clinging to its side. A separate video showed what appeared to be two people falling from a military plane as it flew out of Kabul. 

Since then, crowds have grown at the airport where armed Taliban have urged those without travel documents to go home. At least 12 people have been killed in and around the single runway airfield since Sunday, NATO and Taliban officials said.

(With input from Reuters)