Russian forces are losing ground in territories Putin illegally annexed last week.
Ukrainian forces are pressing on after liberating the strategically important city of Lyman in the Donetsk region on Saturday.
A top DoD official on Monday said Russia could be on track for a "major defeat" in Kherson.
In the four days since Russian President Vladimir Putin announced the illegal annexation of four Ukrainian regions, Russian troops have continued to lose critical ground to Ukraine's armed forces in territories the Kremlin now claims are part of Russia.
Ukrainian forces over the weekend recaptured Lyman, a city of 22,000 in Donetsk — one of the four regions Russia annexed along with Luhansk, Kherson, and Zaporizhzhia.
Lyman was considered a strategically important city for Russian forces. "Lyman was being employed by Russian forces as a logistics hub, so its liberation by Ukraine is a significant operational accomplishment," a senior US military official said during a briefing on Monday.
Ukraine's armed forces are also making significant progress in the southern Kherson region, according to the Pentagon, which Russia hoped to use as a land bridge from the east to Crimea — a peninsula on the Black Sea that Putin illegally annexed in 2014.
Assistant Defense Secretary for International Security Affairs Celeste Wallander on Monday said that Ukraine's gains in Kherson put them on track for what will be a "major defeat" of Russian occupiers — forcing them from Kherson city and to withdraw across the Dneiper river.
"That would be a major defeat for Russia because it pushes back even more Russia's ambition to take Odessa, which was one of the stated objectives earlier this year," Wallander said at a Center for Strategic and International Studies event.
"And it gives Ukraine a much better defensive position to ride out what probably will be a tamping down of the hot fighting over the winter," Wallander said.
Wallander said Ukraine "seems to be on track" to achieve its goals in Kherson, but will need continued support from the US and the international community to get there.
Ukrainian forces made "substantial gains" in Lyman and the Kherson regions between Sunday and Monday, according to an assessment from the Institute for the Study of War (ISW), which is tracking the day-to-day situation on the battlefield in Ukraine.
—ISW (@TheStudyofWar) October 4, 2022
ISW's analysis said that some of the Russian troops who were in the areas recaptured by Ukraine in recent days were "previously lauded as some of Russia's most elite forces."
"The Russian groupings in northern Kherson Oblast and on the Lyman front were largely comprised of units that had been regarded as among Russia's premier conventional fighting forces before the war," ISW said, adding that the apparent failures of these units to hold territory in the face of the Ukrainian counteroffensive bolsters findings that "even the most elite Russian military forces are becoming increasingly degraded as the war continues.
Russian officials bolstered their efforts in Kherson, for instance, with elements of the once-vaunted airborne force known as the VDV, ISW assessed, while the motorized infantry of the 144th Motor Rifle Division was forced to fall back in from Lyman. Heavy losses the 144th suffered heavy losses in September during Ukraine's counter-offensive in Kharkiv and raised speculation it may be combat-ineffective.
Russia's military has suffered staggering losses in the war so far — an estimated 80,000 casualties. Putin in recent weeks also announced a partial mobilization of Russian forces, calling up as many as 300,000 people, some of whom say they are not reservists. Military experts have said the mobilization is unlikely to make a significant short-term difference in Ukraine, given those being drafted have little to no training.
The mobilization has not gone smoothly, with reports of ineligible men being called up as thousands have fled Russia to avoid being drafted, problems that have sparked an amount of dissent rare during Putin's rule. On top of its manpower problems, Russia is also struggling with equipment issues.
"The Russian mobilization system is suffering from severe bureaucratic challenges and limitations that could undermine Putin's efforts to generate the number of troops he needs to continue fighting in Ukraine," ISW said, going to cite Ukrainian military intelligence that suggests "the Russian government is already failing in its promises to properly equip mobilized men and to pay salaries for both mobilized and contract servicemen."
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