UZHGOROD, Ukraine—Vadym Osovsky, a 39-year-old Ukrainian, decided to leave Kyiv with his family on Friday after Russian rockets and paratroopers attacked a nearby military airport and a bridge leading into the city center was destroyed.

Mr. Osovsky drove for more than 50 hours to the border with Slovakia.

Since Ukraine’s government has banned...

UZHGOROD, Ukraine—Vadym Osovsky, a 39-year-old Ukrainian, decided to leave Kyiv with his family on Friday after Russian rockets and paratroopers attacked a nearby military airport and a bridge leading into the city center was destroyed.

Mr. Osovsky drove for more than 50 hours to the border with Slovakia.

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Since Ukraine’s government has banned men under 60 from leaving the country amid a general mobilization aimed at resisting the Russian invasion, Mr. Osovsky said goodbye to his family at the last checkpoint before their passports were stamped.

For several minutes he hugged his wife, two daughters and son, gathering them together for a final photo.

“Don’t worry! We’ll win,” he said as his family walked toward Slovakia.

Vadym Osovsky photographs his family near the Ukrainian border crossing.

Photo: Thomas Grove/The Wall Street Journal

The Osovsky family is among the thousands of people to have fled Kyiv since the Russians invaded Ukraine last Thursday.

Ukraine’s capital of Kyiv has suffered the brunt of the Russian assault on Ukraine, turning swaths of the ancient city of three million into a war zone and forcing thousands of residents to flee to Ukraine’s western borders.

“We were hoping rockets, the artillery would stop, but it only got worse,” said Mr. Osovsky, who moved to Kyiv from an area in eastern Ukraine that had been taken by pro-Russian militants in 2014. “I’ve already seen how this kind of war ends. War can end only very badly.”

Since popular protests that ousted a pro-Russian president a decade ago and set the country on course toward westward integration, Kyiv has become a symbol of Ukraine’s defiance of Russian intimidation.

But Russia’s attacks on the capital city—aimed at decapitating the pro-Western Ukrainian government—have left residents caught between a life-or-death choice between remaining in the city and escaping to safety.

An apartment block in Kyiv which was destroyed by a Russian projectile on Friday.

Photo: Christopher Occhicone for the Wall Street Journal

A destroyed bridge near the town of Bucha in the Kyiv region on Monday.

Photo: MAKSIM LEVIN/REUTERS

Like many among the city’s young, Western-oriented business class, Oleksandra Papadyuk, 40, regularly attended the protests in Kyiv’s Independence Square against efforts by then-President Viktor Yanukovych to bind Ukraine into tighter economic and political relations with Russia.

When those protests turned violent, ousting the widely unpopular president, she rejoiced. For eight years she watched as the city slowly became an Eastern European hub for business and culture.

Ukraine’s second largest city of Kharkiv came under heavy shelling; a nearly 40-mile-long Russian convoy inched closer to Kyiv; President Zelensky addressed the European Parliament. Photo: Sergey Bobok/AFP/Getty Images The Wall Street Journal Interactive Edition

Ms. Papadyuk woke up Thursday morning to the sound of Russian rockets pounding Kyiv’s international airport. She gathered whatever clothes she could find, a toothbrush and some food and left the house in 15 minutes.

“When I was filling up the gas tank, I could feel the ground shake under my feet,” she said.

Her trip to the border with Slovakia took nearly 72 hours by car, waiting for hours with her cousin, her cousin’s nephew and her dog to reach the Uzhgorod border crossing.

Police check vehicles leaving Ukraine at its border with Moldova on Friday.

Photo: Aurel Obreja/Associated Press

A shelter for people fleeing Ukraine in Ubla, Slovakia, on Sunday.

Photo: martin divisek/Shutterstock

“For the first two hours, I was in complete denial. I told myself I was going for a car ride,” Ms. Papadyuk said, fighting tears. “I never thought Russia could do this.”

For many, even after leaving the city, the trip was fraught with danger. Abandoned cars littered the median strip of the main roads out of Kyiv after breaking down or running out of gas. People were standing on the side of the road trying to hitch a ride west, said others who made the trip.

Amid the jam of vehicles leaving the city center, traffic moved at five to 10 miles an hour for long parts of the day. Lines at gas stations snaked hundreds of meters along the road, and motorists sped west down eastward-bound lanes, dodging armored vehicles and trucks hauling surface-to-air rockets toward the city. Shops were closed, and people rode bikes or walked on the roads leading west.

Since the first wave of evacuees left the city, violence has intensified in Kyiv. Over the weekend an explosion carved a crater in the ground at the base of a 10-floor apartment block and blasted through scores of apartments, blowing out windows on either side. The windows of a kindergarten 50 yards from the blast were also shattered.

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Manel Khaled, 23, who moved to Kyiv from Morocco to become a pharmacist, said that on Saturday morning, she decided to flee.

“At one point, they stopped just attacking military targets, but civilian targets, like right next to our dormitory,” she said. “My heart aches for the people staying there.”

Olena Sturrock, who left Kyiv before the violence began, said her mother is still in the city and that she has tired of going up and down the stairs to the bomb shelter in the basement of her building.

She talks to her daily by phone. But over the weekend her mother said a last goodbye, just in case.

“I love you very much,” her mother said. Ms. Sturrock told her she will be fine and they will all be together when it is over.

Write to Thomas Grove at thomas.grove@wsj.com