With the furious battle for the city of Bakhmut raging at their backs, a squad of Ukrainian soldiers tore through an open field, racing to get out of range of falling Russian artillery.

But before they could make it to safety, they said, they got a flat tire.

The three soldiers — known by the call signs Omar, Chip and Bandit — had spent the day on Friday taking part in Ukrainian offensive operations on the edge of Bakhmut in eastern Ukraine, blasting Russian tanks and armored vehicles. But after surviving another brutal day of battle, they worried that the punctured tire might doom them.

Omar, 36, hopped out of the car and used a screwdriver to put a plug in the hole. Within moments, they were off again.

The men, who recounted their story on Saturday just outside the nearby town of Chasiv Yar, were among the hundreds of Ukrainian soldiers ordered on the offensive around Bakhmut in recent days. The fighting has often been hard, they said, with many Russians willing to die rather than surrender even when surrounded.

But every step forward, they said, was better than a step back.

For nearly a year, Ukraine has been simply trying to hold on in Bakhmut as Russian forces pressed in on the city from both sides while at the same time laying waste — block by bloody block — to what had once been a vibrant coal-mining city of 80,000.

Over time, Bakhmut has taken on an outsize importance: a symbol of Ukrainian defiance and of Russian leaders’ determination to blast their way to a small victory in a little-known corner of eastern Ukraine.

Last week, for the first time, Ukrainian forces launched a series of coordinated counterattacks and in a matter of days won back territory north and south of the city that it had taken Russian forces months to capture. Ukrainian soldiers hope they have now turned the tide in the battle and can continue moving forward.

“Everything is better when we start kicking these Russians out,” said Omar, who asked to be identified only by his call sign in accordance with military protocol.

But inside Bakhmut, soldiers fighting to hold onto the last few destroyed buildings they control in the western corner of the ruined city said on Saturday that it remained a place of incomprehensible violence.

“It is too early to talk about successes in the city itself,” said a member of a platoon rotating out of Bakhmut. He was one of 16 soldiers waiting for a military transport in Chasiv Yar after a monthlong tour. “We need more Javelins,” he said, referring to the U.S.-made missiles supplied to Ukraine. “Russian tanks are giving us nightmares.”

Chasiv Yar, about six miles to the west of Bakhmut, is itself under siege as the two sides trade volleys of artillery and rockets. It is now the next line of defense for Ukraine if the Russians were to take Bakhmut and continue advancing west.

“Our troops are gradually advancing in two directions in the suburbs of Bakhmut,” Hanna Mailar, a Ukrainian deputy minister of defense, said in a statement on Saturday night. She said active hostilities made it impossible to give precise details about the state of the fighting but claimed that the Ukrainians “are destroying the enemy and have already taken many prisoners.”

“The situation in the city itself is more complicated,” she said.

Since December alone, the United States estimates that more than 20,000 Russian soldiers have been killed in fighting in Ukraine, many around Bakhmut, the National Security Council spokesman, John Kirby, said this month. A further 80,000 have been wounded, he said.

The Ukrainians are now seeking to exploit those losses by attacking the positions around the city that they lost in the winter and trying to encircle the Russian forces inside the ruins of Bakhmut.

Russian forces control more than 90 percent of the city, according to Russian and Ukrainian officials, but the Ukrainians are keeping them from taking the last bit of land and advancing on Chasiv Yar.

Under the shelter of a battered bus stop in Chasiv Yar, a platoon of soldiers rotating out of Bakhmut did not flinch as outgoing artillery boomed and incoming rockets thundered close by. They had faced much worse.

The commander of the 24th Motorized Rifle Brigade, who goes by the call sign Prince, said on Saturday that after taking a short tactical pause, Russian forces were furiously assaulting the city again.

“Artillery fire, rocket and airstrikes do not stop for a minute,” he said. “Every meter of the city is now under shelling.”

The brigade released a photo of the devastation taken by a drone on Saturday that they said captured the annihilation of Bakhmut. The outlines of foundations of what used to be homes and businesses were shrouded in a soupy fog that the brigade said was smoke from fires and shelling. The authenticity of the photo could not be independently confirmed, but it tracks with reports from those who have been in the devastated city.

For the last week, sometimes twice a night, the Ukrainian soldiers said, Russians have been using incendiary munitions to force them from their positions.

The Ukrainians claim that the bombs are phosphorus munitions, which are banned under international law when used against civilians but not when used against military targets, and can burn at temperatures over 1,400 Fahrenheit.

The smell of fire and smoke hung heavy in Chasiv Yar on Saturday as the shellfire shook the valley below. There were fresh impact craters on streets around the town.

There were only a handful of civilians, mostly older people, on the streets. They went about their business as a steady stream of armored vehicles and tanks rumbled past, heading into battle.

“We feel so bad for them,” Marcello, one of the soldiers, said of the few remaining civilians in Bakhmut. “There was a 92-year-old babushka who was injured, and we put her on the stretcher and helped get her out.”

As they waited for their own ride to safer ground, they were just learning that the Russians outside that city were reported to be suffering a series of defeats.

“Finally,” one soldier said.