ANDRIIVKA, Ukraine — The men of Ukraine’s 37th brigade were freshly trained and armed with Western-supplied weapons, tasked with an initial push through Russian-occupied territory in the early days of a long-awaited counteroffensive.

They would pay a heavy price.

Within 20 minutes of their June 5 advance south of Velyka Novosilka, in the southeast Donetsk region, mortars exploded all around them, soldiers said. A 30-year-old soldier known as Lumberjack saw two of the men in his vehicle bleeding heavily; one lost an arm as he cried out for his family. Lumberjack crawled into a crater, but the shrapnel from a mortar went through the soil and pierced his shoulder.

“We were left there in the field, without tanks or heavy armor,” said Lumberjack, who spoke to The Washington Post on the condition that he be identified only by his call sign because he was not authorized to discuss the battle. “We were shelled with mortars from three sides. We couldn’t do anything.”

There were fewer than 50 men in the unit, he said, and 30 did not return — they were killed, wounded or captured by the enemy. Five of the unit’s armored vehicles were destroyed within the first hour.

The 37th’s reconnaissance push would eventually help other Ukrainian brigades liberate four villages in Donetsk, one of the counteroffensive’s key starting points, securing an early triumph for Kyiv during the most widely anticipated operation of the war.

Survivors from the 37th, which included some American volunteer fighters and soldiers trained in Europe, described their ordeal as the tip of the spear in Ukraine’s counteroffensive.

Their accounts offered one of the first real glimpses into the brutal fighting on a new front this month, far removed from the antiseptic claims of advances and retreats by officials on both sides of the conflict. And the battalion’s heavy losses foreshadow the terrible cost that Ukraine’s leader are prepared to pay — and believe they must pay — to oust the Russian invaders and reclaim their lost territory.

Elsewhere on Tuesday, Ukrainian units pressed the fight in other key areas along the front: in the Donetsk region — east of Lyman, on the outskirts of Bakhmut, and in Marinka — and in the Zaporizhzhia region, south of Orikhiv.

“The enemy is doing everything to keep the positions [they] captured,” Ukrainian Deputy Defense Minister Hanna Maliar wrote in a Telegram post, adding that Russians were fighting back with “assault and army aviation” and “intense artillery fire.”

Russia responded to Kyiv’s counteroffensive with its latest deadly overnight missile barrage, killing 11 people and injuring at least 36 in President Volodymyr Zelenskyy’s hometown of Kryvyi Rih.

Cruise missiles struck a residential building and a warehouse in the working-class city in central Ukraine.

Zelenskyy posted a video on his Telegram channel, showing flames leaping from the ruins of a building and the smoldering remains of cars. “More terrorist missiles,” Zelenskyy wrote. “Russian killers continue their war against residential buildings, ordinary cities and people.”

Just hours after the strike, Russian President Vladimir Putin denied that his country hits civilian targets. In a wildly cynical claim to pro-war bloggers and military journalists he accused Ukraine of being the aggressor in a war that he, in fact, brazenly initiated more than a year ago. “They started this war,” he said, “and we are trying to stop it.”

“The enemy has had no success in any of the areas,” Putin said, adding: “During this time, Ukraine has lost over 160 tanks and over 360 armored vehicles. . . . Our losses are 10 times less.” His assertions could not be confirmed.

On Tuesday, the Biden administration said it would send Ukraine an additional $325 million in military supplies from U.S. stockpiles. The package includes 25 armored vehicles, possibly to replace those destroyed as fighting intensifies, plus additional Javelin missile launchers and other weapons for targeting Russian tanks, more artillery and air defense munitions, and millions of small-arms rounds and grenades.

In a meeting with President Biden at the White House on Tuesday, NATO Secretary General Jens Stoltenberg thanked the United States for the new aid package and said the Ukrainian counteroffensive was “making progress.”

“The more land the Ukrainians are able to liberate, the stronger hand they will have at the negotiating table,” he said.

Ukrainian forces, meanwhile, continued to push ahead on a critical southeast front line here in the vast farmlands and fields of the Donetsk region.

Kyiv claims to have liberated four villages — Neskuchne, Storozheve, Blahodatne and Makarivka — just south of Velyka Novosilka, on a roughly 5½-mile stretch of road close to the border of the Zaporizhzhia region and about 80 miles north of the decimated, Russian-occupied city of Mariupol.

Serhii Kozachynskyi, a spokesman for the 35th separate marine brigade, said his unit helped clear two of the villages without any of the newly supplied weapons and equipment from Ukraine’s Western allies, which are boosting other brigades along the front line. While some of his fellow soldiers had trained in countries including Britain, Norway and Spain, the entire brigade had not been sent abroad, as was the case with some specialized attack units.

His brigade cleared the village of Storozheve on Sunday morning and Makarivka on Monday, Kozachynskyi said. Eighty to 90 percent of the homes in both villages were destroyed. At least 17 people were evacuated from the four liberated villages, a Donetsk police press officer said.

“The soldiers are highly motivated to move the enemy out of our territory, to finish the war as fast as they can,” Kozachynskyi said. “Even not having all of the equipment, they can perform the tasks that are required.”

Sometimes, even the Western-supplied gear hasn’t been enough to shield outmatched Ukrainian troops from the intensity of Russian shelling.

For the first hour and a half of the 37th’s assault near Velyka Novosilka, the Russians bombarded the unit with nonstop shelling that penetrated their AMX-10 RC armored vehicles, according to Grey, another soldier in the battalion who spoke on the condition that he be identified only by his call sign. The armored vehicles, sometimes called “light tanks,” were not heavy enough to protect the soldiers, Grey said, and had to be positioned behind them instead of in front.

As mortars barreled toward them, Lumberjack crawled out of his vehicle for more than 300 yards, he said, desperately searching for the tree line.

“Everyone expected that we would have some kind of support, but unfortunately, for some reason, there was none,” Lumberjack said. His commander had little experience, he said, and had counted on assistance from artillery units. “But he got confused when he saw that there was none.”

Finally, after two hours of shelling, the Russians appeared to run out of ammunition, or were exhausted by the fight. Ukrainian forces seized the moment, Lumberjack recounted, taking the positions of their enemies and clawing back some ground.

The mission, brutal as it was, distracted Russian forces, allowing Ukraine to attack on other flanks. After the initial push of the 37th, other brigades helped liberate the four villages.

“Yes, we made a sacrifice,” he said. “But we got a result in return.”

Unlike most of the men in his unit, Lumberjack’s injury was relatively minor. But the shrapnel was still lodged in his shoulder, making it difficult for him to fire a weapon, so he planned to head to Kyiv for treatment.

But first, for a brief moment Tuesday afternoon, Lumberjack and Grey stopped at a creek in a town just north of those they helped liberate. Soldiers jumped into the water and swam beneath the summertime sun. They hardly seemed to notice the sounds of cluster munitions and artillery nearby.

Grey planned to return to the fight that night, to replace weary soldiers in the counterattack.

Their goal was clear: Retaking all of Ukraine’s territory from Russia. Their next big target? Mariupol, they said.

“We have to keep pushing forward,” Grey said. “We are afraid of death but we understand we don’t have another choice. We need to go and push the enemy away from here.”

The Washington Post’s David L. Stern in Mukachevo, Ukraine; Francesca Ebel in London; and Dan Lamothe and Matt Viser in Washington contributed to this report.

previous coverage

• US to send Ukraine more Strykers and Bradleys in new $325M aid package
• The potent US arsenal for Ukraine’s counteroffensive
• Pentagon announces new long-term arms package for Ukraine worth up to $2.1 billion