Hundreds of Afghans picked their way on Friday through the carnage left behind by the previous day’s deadly suicide bombings outside Kabul airport in a last-ditch effort to flee the country, as the U.S. and its allies rushed to complete their final evacuation flights.

The official Afghan death toll from the attack, claimed by Islamic State, was expected to rise above the current figure of 90. One hospital alone, the Wazir Akbar Khan Hospital in central Kabul, said it had received the bodies of 145 people killed.

The U.S., which lost 13 members of its armed services, vowed a retaliatory strike against Islamic State’s local offshoot, known as ISIS-K.

On Friday, the Taliban, which has been manning checkpoints around the airport, criticized Washington for lax security that it said opened the way for the bombers. The Taliban have for years fought Islamic State as the two Islamist militant groups have battled for supremacy in Afghanistan.

“The incident did not happen in an area controlled by the Islamic Emirate,” as the Taliban refer to themselves, said Habib Samangani, a member of the Taliban’s cultural commission. “That area is controlled by the Americans. We blame the Americans for it.”

Mr. Samangani said Taliban intelligence officials were investigating the attack, but that the probe was still at a “preliminary stage.”

A Taliban fighter stood guard Friday at the site of the twin suicide bombings.

Photo: wakil kohsar/Agence France-Presse/Getty Images

Marine Corps Gen. Frank McKenzie said in a press briefing Thursday after the explosions that the threat from Islamic State, along with “other active threat streams,” remained.

“We believe it is their desire to continue these attacks and we expect those attacks to continue,” he said.

Thursday’s violence appears to have sped up the pace of evacuations from Kabul airport, with the U.K. saying it would finish flights Friday morning. Other countries expressed regret that they wouldn’t be able to airlift all citizens and Afghans who worked with their forces and embassies.

As crowds queued up in the street outside Hamid Karzai International Airport in the early morning, dozens of Afghan men waded through open sewers lining the street to get closer to the gates.

“There was still blood and pieces of flesh and torn-off clothes on the ground,” said a 29-year-old man who lives near Kabul airport. “People stepped on it trying to get to the airport.”

The crowd was smaller than Thursday, with many apparently wary in the wake of the attack.

Later Friday, rumors spread of another explosion, sending people running away from the airport in all directions, leaving only Taliban fighters guarding the gates, according to a shopkeeper in the area. Taliban militants have since prevented would-be evacuees from getting near the airport, several witnesses said.

President Biden in a speech Thursday evening said he had instructed his military commanders to develop response plans to the attack.

“We will hunt you down and make you pay,” Mr. Biden said, adding that U.S. evacuations would continue despite the attack, and that efforts to extract Americans who want out of Afghanistan would continue beyond an Aug. 31 deadline he set for withdrawing the last remaining American troops.

A relative mourns next to the body of a blast victim at a hospital in Kabul.

Photo: stringer/Shutterstock

Meanwhile, U.S. officials and activists stepped up efforts to get as many Westerners and Afghans out of the country as they could. In recent days, access to the airport had been impeded by Taliban checkpoints and bureaucracy at the airport, leaving several evacuation flights to take off with significant numbers of empty seats.

Americans held tense negotiations with Taliban leaders to get approval to bring busloads of Afghans to the airport for flights to places like Ukraine and Albania.

But the dangers at the airport also pushed Western organizers to turn toward other escape routes. More people trying to flee Taliban rule turned to the roads to Pakistan, which was preparing to accept a new surge of Afghan refugees from its neighbor.

The U.K. finished processing documents for about 1,000 people it wanted to evacuate by 4.30 a.m. Friday, the country’s defense secretary, Ben Wallace, said in a broadcast interview.

“The main processing has closed and we have a matter of hours,” he said. “We will continue to honor our debt to all those who have not yet been able to leave Afghanistan. We will do all that we can to ensure they reach safety.”

Australia and New Zealand said Friday that they had flown their last evacuation flight from Afghanistan, due to terrorist threats.

Australian Prime Minister Scott Morrison said his country’s military personnel had been evacuated from Kabul just hours before the attacks and that the precarious security situation made further evacuations unsafe.

“The devastating thing is that we weren’t able to bring everyone, and now we need to look to see what we can do for those who remain,” New Zealand Prime Minister Jacinda Ardern said in a news conference Friday.

Meanwhile, Turkish President Recep Tayyip Erdogan said Friday that Ankara was still considering a request from the Taliban to provide civilian technical assistance to operate Kabul airport.

The Taliban request of Turkey—a Muslim-majority member of the North Atlantic Treaty Organization that backed the fallen Afghan republic—underlines the Islamist movement’s need for assistance as it seeks to govern the country.

While all Western nations have closed down their embassies and airlifted staff out of the country, Russia, which has backed the Taliban as a guarantor of stability for Afghanistan, has so far appeared determined to keep its embassy in Kabul running, but has evacuated several hundred citizens.

Russia’s ambassador to Afghanistan, Dmitry Zhirnov, told Russian television that the Taliban were attentive to security at the embassy, where he said work was continuing smoothly.

—Alan Cullison, Zamir Saar, Jalaluddin Nazari, Ehsanullah Amiri and Dion Nissenbaum contributed to this article.

Write to Sune Engel Rasmussen at sune.rasmussen@wsj.com