The United Nations humanitarian chief said early Friday that he was on his way to visit parts of Turkey and Syria stricken by a powerful earthquake earlier this week, hours after the death toll climbed past 20,000.

The official, Martin Griffiths, will visit Aleppo and Damascus in Syria, along with Gaziantep in Turkey, over the weekend, according to the U.N. secretary general, António Guterres.

“More help is on the way, but much more, much more is needed,” Mr. Guterres told reporters on Thursday.

Earlier in the day, the toll from Monday’s 7.8 magnitude earthquake surpassed the one in 1999 that struck about 60 miles from Istanbul and killed about 17,000 people, according to official figures. Monday’s quake is now Turkey’s deadliest since 1939 and one of the deadliest worldwide in decades.

Turkey imposed a three-month state of emergency Thursday in 10 provinces affected by the quake. In many of the hardest-hit areas, a chaotic atmosphere prevails on the ground.

In the city of Antakya, whose ancient old town has entirely collapsed, traffic has been bumper-to-bumper, the air is acrid from the smoke of bonfires as people try to stay warm, and ambulances and trucks bringing aid arrive nonstop. Thousands of people have been sheltering in white tents in the shadow of a soccer stadium.

As the humanitarian crisis continues, the possibility of further seismic activity hangs over the region. Dozens of earthquakes have been recorded in Turkey this week, including a few on Friday morning. Experts say that large aftershocks, like the ones that hit within hours of the initial temblor, pose potential risks to the structural integrity of partially collapsed structures in the earthquake zone.

The effort to provide aid to millions of people has also been strained by bitter cold, power outages and shortages of fuel, trucks and other essential supplies.

Relief organizations say that the first 72 hours after a natural disaster are crucial for finding survivors. Once that window passes, as it already has in Turkey and Syria, the medical burden typically shifts from disaster sites to medical facilities, said Yasushi Nakajima, an expert on disaster risk management at Tokyo Metropolitan Hiroo Hospital in Japan.

“On the other hand, medical facilities that have been responding to the disaster since it began will face major problems of staff fatigue, supply shortages, and supply disruptions, including fuel and water,” Dr. Nakajima said.

“Lives that could have been saved under a normal medical system are being lost as if sand were slipping from their hands,” he added.

Syria, where millions have been displaced by a yearslong civil war, faces its own distinct humanitarian challenges in getting aid.

There have been a few breakthroughs so far. Six humanitarian aid trucks carrying shelter material and nonfood items entered the opposition-held region of northwestern Syria on Thursday, marking the first significant delivery of aid to the area since the earthquake. The International Organization for Migration said that items in the convoy could meet the needs of at least 5,000 people.

But Syria has received far less aid than Turkey since the earthquake, in part because the border crossing used on Thursday, known as Bab al-Hawa, is the only one to allow United Nations aid to reach the opposition-held region in Syria’s northwest.

Mr. Guterres said that the greatest obstacle for delivering aid to Syria was access.

The United Nations will ask the Security Council for authorization to expand aid to Syria, a U.N. spokesman, Stéphane Dujarric, said, adding that the U.N. was also negotiating with Syria’s government to deliver more humanitarian aid to opposition-held areas in the northwest.

Syria’s government has said that United States sanctions against the country have exacerbated the humanitarian disaster caused by the earthquake. Those sanctions do not target humanitarian aid.

On Thursday, the U.S. State Department doubled down on its refusal to lift sanctions on Syria, insisting that humanitarian aid efforts were not impeded by the policy. It also repeated a demand that President Bashar al-Assad’s authoritarian government open more border crossings for aid delivery.