Military personnel fired tear gas to control the crowds of Afghans trying to gain access to Kabul airport Friday, a day after the Pentagon said that order was being restored at the site and evacuation flights from Afghanistan would be accelerating.

Soldiers have also fired into the air to disperse the crowd, according to a senior Western official. It was unclear as to whether the soldiers were American; British, Afghan and other Western troops are also stationed at the airport.

There was no immediate comment from the U.S. military.

Soldiers are also going just outside the airport perimeter, which is surrounded by Taliban, to disperse crowds and clear the way for families struggling to get in, videos clips taken by passengers show.

Thousands of Afghans are still pushing to get into the airport, following the Taliban’s takeover of Kabul on Sunday. North Atlantic Treaty Organization allies and partners have evacuated nearly 20,000 people from Kabul airport since last weekend, according to public announcements by officials.

To reach the airport, Afghans and foreigners have to get past Taliban checkpoints, where Taliban fighters are firing in the air and using violence to hold back crowds.

Chaos continued around Kabul’s international airport as evacuations out of Afghanistan remained difficult for many, including Afghan women, who face an uncertain future under Taliban rule. WSJ’s Jessica Donati explains. Photo: Isaiah Campbell/U.S. Marine Corps/AFP/Getty Images The Wall Street Journal Interactive Edition

The three entry gates to the airport remain blocked with people pushing to get inside. In order to allow families, including women and children, into the airport, security forces are using tear gas and firing into the air to disperse the crowd, Western officials said.

Tear gas was used Thursday and Friday at the airport, say people who have accessed the airport, and officials.

A senior British official said Friday morning that the government was speeding up evacuations and had flown 963 people from Kabul in the past 24 hours.

“That’s a significant acceleration on the numbers over the previous 24 hours,” Armed Forces Minister James Heappey told Sky News, “and reflects I think that we’re now fully up to speed, with military and foreign office consular staff all now up and running.”

The numbers evacuated in the coming days should be similar, he said.

France has sent three transport planes to Kabul that have airlifted around 500 people out of the country. Paris has also deployed special forces in Afghanistan to escort people to the airport. Those operations rely heavily on cooperation with U.S. authorities that control the airport, a French official said. “It’s not a question of willingness for us, but of feasibility,” the official said.

Mr. Heappey said that he understood that the Taliban weren’t turning people away from the airport.

“Where they have done that, I’ve heard that it’s more that they are being officious rather than malicious,” he said. People called for flights should have confidence that the Taliban would let them through, he said.

Mr. Heappey said he couldn’t put a time frame on how long the Royal Air Force’s evacuation operation would last.

“Eventually the air bridge will have to close and quite possibly not everybody will have been got out. That’s what keeps us awake at night. That’s what’s motivating to us to work as hard as we are to ensure that those numbers are the absolute minimum,” he said.

Getting people through the city of Kabul, past Taliban checkpoints, and inside the gates of the airport has proven immensely difficult, German military officials said.

Germany is sending two small transport helicopters to Kabul to assist in extracting people from Kabul to transport them to the airport, German Defense Minister Annegret Kramp-Karrenbauer told reporters Friday. The helicopters are specially designed for extraction in tight spaces and are expected to be operational in Kabul on Saturday.

The deployment of the German military helicopters was agreed to on Wednesday this week and won’t be restricted to extracting German nationals or Afghan civilians, Mrs. Kramp-Karrenbauer said.

German Inspector General Eberhard Zorn told reporters that the helicopters would only be deployed in extraordinary circumstances, but didn’t elaborate. The helicopters would be armed and fly with support from larger U.S. machines too big to land in the city.

“This is not a taxi service,” Gen. Zorn told reporters.

A German national was shot in unclear circumstances while traveling to the Kabul airport, German officials said Friday. The man, a civilian whose name is being withheld by authorities, was treated for wounds that weren’t life-threatening and would be evacuated.

One Afghan man who has worked for a U.S. contractor and who has an approved Special Immigrant Visa for the U.S. said he arrived at the airport on Thursday and attempted all night to enter. His family carried their children on their shoulders to prevent them from being crushed. However, he failed and gave up Friday morning, returning home exhausted.

“We were stuck between the aggression of the Taliban and U.S. forces in the gate,” said the man. “I don’t know if I will ever be able to get out.”

In addition to the thousands of American troops at the airport, including Marines, there are British soldiers as well as soldiers from Afghan security forces who didn’t surrender to the Taliban and who are also providing security in cooperation with international troops. These Afghans are members of CIA-trained units, Western officials say.

At the checkpoints, the Taliban are searching for key individuals from the ousted government, a report from the Norwegian Center for Global Analyses said. The Taliban are going house to house in Kabul, hunting down people associated with the former government of President Ashraf Ghani, the Afghan leader who fled the country on Sunday.

If the militants can’t find the individuals they seek, they are arresting members of their families, the report said. Before taking Kabul and other major cities, the group had already assembled a target list of individuals, which has now been augmented.

“The Taliban are arresting and/or threatening to kill or arrest family members of target individuals unless they surrender themselves to the Taliban,” the report said. “Taliban are rapidly recruiting new informer networks eager to collaborate with the new regime.”

—James Marson contributed to this article.

Write to Saeed Shah at saeed.shah@wsj.com