A powerful earthquake shook Nepal on Saturday near its capital, Katmandu, killing more than 1,300 people, flattening sections of the city’s historic center, and trapping dozens of sightseers in a 200-foot watchtower that came crashing down into a pile of bricks.

Officials in Nepal put the preliminary number of deaths at 1,360, nearly all of them in Katmandu and the surrounding valley, with 4,279 people injured. But the quake touched a vast expanse of the subcontinent. It set off avalanches around Mount Everest, where at least 10 climbers died. At least 34 deaths occurred in northern India. Buildings swayed in Tibet and Bangladesh.

The earthquake, with a magnitude of 7.8, struck shortly before noon, and residents of Katmandu ran into the streets and other open spaces as buildings fell, throwing up clouds of dust. Wide cracks opened on paved streets and in the walls of city buildings. Motorcycles tipped over and slid off the edge of a highway.

Seismologists have expected a major earthquake in western Nepal, where there is pent-up pressure from the grinding between tectonic plates, the northern Eurasian plate and the up-thrusting Indian plate. Still, witnesses described a chaotic rescue effort during the first hours after the quake as emergency workers and volunteers grabbed tools and bulldozers from construction sites, and dug with hacksaws, mangled reinforcing bars and their hands.

Though many have worried about the stability of the concrete high-rises that have been hastily erected in Katmandu, the most terrible damage on Saturday was to the oldest part of the city, which is studded with temples and palaces made of wood and unmortared brick.

Four of the area’s seven Unesco World Heritage sites were severely damaged in the earthquake: Bhaktapur Durbar Square, a temple complex built in the shape of a conch shell; Patan Durbar Square, which dates to the third century; Basantapur Durbar Square, which was the residence of Nepal’s royal family until the 19th century; and the Boudhanath Stupa, one of the oldest Buddhist monuments in the Himalayas.

For many, the most breathtaking architectural loss was the nine-story Dharahara Tower, which was built in 1832 on the orders of the queen. The tower had recently reopened to the public, and visitors could ascend a narrow spiral staircase to a viewing platform around 200 feet above the city.

The walls were brick, around one and a half feet thick, and when the earthquake struck, they came crashing down.

The police said on Saturday that they had pulled around 60 bodies from the rubble of the tower. Kashish Das Shrestha, a photographer and writer, spent much of the day in the old city, but said he still had trouble grasping that the tower was gone.

“I was here yesterday, I was here the day before yesterday, and it was there,” he said. “Today it’s just gone. Last night, from my terrace, I was looking at the tower. And today I was at the tower — and there is no tower.”

Kanak Mani Dixit, a Nepalese political commentator, said he had been having lunch with his parents when the quake struck. The rolling was so intense and sustained that he had trouble getting to his feet, he said. He helped his father and an elderly neighbor to safety in the garden outside and then had to carry his elderly mother.

“And I had time to do all that while the quake was still going on,” Mr. Dixit said. “It was like being on a boat in heavy seas.”

Roger Bilham, a professor of geological sciences at the University of Colorado, said the shaking lasted about one minute, although it continued for another minute in some places.

Joydeb Chakravarty, managing director of the J. Walter Thompson advertising agency in Nepal, said he had been grocery shopping when the quake struck. “And suddenly, everything started collapsing around us,” he said. “The shelves all came down, the food items all crashed down. We were barely able to get out the emergency exit.”

For years, people have worried about an earthquake of this magnitude in western Nepal. Many feared that an immense death toll would result, in part because construction has been largely unregulated in recent years, said Ganesh K. Bhattari, a Nepalese expert on earthquakes, now living in Denmark.

He said the government had made some buildings more robust and reinforced vulnerable ones, but many larger buildings, like hospitals and old-age homes, remained extremely vulnerable. “There is a little bit of improvement,” he said. “But it is really difficult for people to implement the rules and the regulations.”

Saturday’s earthquake struck when schools were not in session, which may have reduced the death toll. But there was not yet a full picture of the damage to villages on the mountain ridges around Katmandu, where families live in houses made of mud and thatch.

As night fell, aftershocks were still hitting, prompting waves of screaming. Many residents sat on roads for much of the day, afraid to go back indoors, and many insisted that they would spend the night outside despite the cold. Thousands camped out at the city’s parade ground. The city’s shops were running short of bottled water, dry food and telephone charge cards.

Toward evening, hospitals were trying to accommodate a huge influx of patients, some with amputated limbs, and were running short of supplies like bandages and trauma kits, said Jamie McGoldrick, resident coordinator with the United Nations Development Program in Nepal. Water supplies, a problem under normal circumstances in this fast-growing city, will almost certainly run short in the coming days, he said. Search and rescue personnel will face the challenge of reaching villages nearer the earthquake’s epicenter, about 50 miles northwest of Katmandu, where damage may be catastrophic.

Secretary of State John Kerry said the American ambassador to Nepal, Peter W. Bodde, had issued a disaster declaration that would allow $1 million in humanitarian assistance to be available immediately. A disaster response team and an urban search-and-rescue team from the United States Agency for International Development will also be deployed, he said in a statement,

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China and India, which jockey for influence in the region, have pledged disaster assistance.

The earthquake set off avalanches on Mount Everest, where several hundred trekkers were attempting an ascent, according to climbers there. Alex Gavan, a hiker at base camp, called it a “huge disaster” on Twitter and described “running for life from my tent.” Nima Namgyal Sherpa, a tour guide at base camp, said in a Facebook post that many camps had been destroyed.

At least 10 people died on Mount Everest after the earthquake, Nepalese officials said.

Tremors from the quake were felt across northern India, rattling bookcases and light fixtures as far away as New Delhi. Electricity was switched off for safety reasons in the Indian state of Bihar, where three deaths were reported in one district, Rajiv Pratap Rudy, India’s minister of skill development, told reporters in New Delhi. Two deaths were reported in another district.

The region has been the site of the largest earthquakes in the Himalayas, including a 2005 quake in the Kashmir region and a 1905 earthquake in Kangra, India.