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"The situation (in Bakhmut) is quite difficult," Col. Gen. Oleksandr Syrskyi, the Ukrainian commander of ground forces, said in a statement on the Telegram messaging app.
"At the same time, in certain parts of the city, the enemy was counterattacked by our units and left some positions" in recent days, he said.
The 10-month-long battle for the eastern Ukrainian city has taken on a symbolic importance for both sides. It has become the fulcrum of a war that has seen little shift in front lines since late 2022, leaving both sides looking for a breakthrough.
On Monday, Russia unleashed a fresh volley of missiles on Ukraine overnight that killed two people in the east, set off huge blazes and damaged dozens of homes and other buildings.
The White House said on Monday that Russia has exhausted its military stockpiles and its armed forces with some 100,000 Russian troops killed or wounded in Ukraine in the past five months. Of the 20,000 killed, half have been from the Wagner Group.
"Russia's attempt at a winter offensive in the Donbas largely through Bakhmut has failed," Kirby said.
He said the Russians have made some incremental gains in Bakhmut but that this has come at a "terrible, terrible cost" and that Ukraine's defenses in the region remain strong.
"Russia has exhausted its military stockpiles and its armed forces," Kirby said.
Most of the Wagner mercenary group's soldiers were "Russian convicts thrown into combat in Bakhmut without sufficient combat or training, combat leadership, or any sense of organizational command and control", he said.
"It's really stunning, these numbers," Kirby added, saying the total is three times the number of American casualties in the Guadalcanal campaign in World War Two.
Kirby said another US weapons package for Ukraine would be announced soon.
Yevgeny Prigozhin, head of Wagner, said on Telegram on Monday that his fighters needed some 300 tonnes of artillery shells a day for the assault on Bakhmut, but were receiving only a third of that amount.
"Three hundred tonnes a day is 10 cargo containers - not a lot at all," said Prigozhin, who has often clashed with Russia's defence establishment over its conduct of the war in Ukraine.
In a separate posting on Monday evening, Prigozhin said his troops had advanced some 120 metres into Bakhmut at the loss of 86 of his fighters.
Syrskyi said that new Russian units are being "constantly thrown into battle for Bakhmut" despite taking heavy losses, Syrskyi said, adding: "But the enemy is unable to take control of the city."
Russian forces have steadily made incremental gains in Bakhmut, but Ukraine said on Sunday that it was still possible to supply the defenders with food, ammunition and medicine.
Kyiv is widely expected soon to launch a counter-offensive to retake swathes of territory in the east and south that were occupied by Russian forces following the invasion 15 months ago.