It was not immediately possible to confirm either side’s account of the fighting.

Though Ukraine’s battle plans target specific areas for trying to break through Russian lines, its troops are testing Russian defenses in many places for weaknesses and poor morale, and are prepared to shift position to concentrate in those areas.

The most recent battle erupted in an area of table-flat steppe in Ukraine’s south, an unforgiving landscape for war with little cover for advancing troops, where Russia has laid mine belts and dug layer upon layer of defensive trenches. Ukraine’s military and a local Russian-appointed official, Vladimir Rogov, said on Friday that heavy fighting was taking place south of the town of Orikhiv.

That area is near where military analysts have said for months that the Ukrainians might focus the brunt of a counteroffensive operation, armed with their newly acquired arsenal of Western tanks and armored personnel carriers. Kyiv’s push there, analysts say, is part of an effort to drive a wedge into land that Russia has seized since launching its full-scale invasion last year, splitting the territory into two and cutting Moscow’s supply lines to the occupied Crimean Peninsula.

Beginning on Thursday, Russian pro-war bloggers and the Russian military reported that Ukraine had unsuccessfully attempted an advance a few miles east of Orikhiv, near the village of Mala Tokmachka. Videos and photos posted by the bloggers, and verified by The New York Times, show that at least three Leopard 2 tanks and eight American-made Bradley fighting vehicles were recently abandoned by Ukrainian troops or destroyed.

Two U.S. officials, speaking on condition of anonymity to discuss sensitive military operations, said that advancing Ukrainian forces had suffered casualties, but that American assessments quantifying those losses were still being developed. The officials said that Russians had also suffered some casualties.

In most military campaigns, attacking forces typically suffer higher initial losses than dug-in defenders, and much of the terrain in Ukraine’s south is flat, open ground — leaving troops and armored vehicles exposed to enemy artillery.

The Times also verified, according to images posted by a Ukrainian brigade, that Ukrainian troops were on foot in Lobkove, a settlement west of Orikhiv, for the first time in the vicinity of the offensive. But the videos and images have emerged hours or days after they were taken, and many have come from pro-Russian sources, making it impossible to determine which side has an advantage in the fighting.

About 60 miles east of Orikhiv, Ukraine was also attacking across the plains near the town of Velyka Novosilka, where a deputy defense minister, Hanna Malyar, said on Thursday that a battle was being fought. Ukrainian officials have also said they are attacking Russian positions on the outskirts of the eastern city of Bakhmut, which fell to Russian forces last month after the longest and bloodiest battle of the war. Ukraine’s military said on Thursday that it had advanced by several hundred yards near Bakhmut.

The attack near Orikhiv is plowing directly into a dense line of Russian defenses, suggesting that Ukraine might try a direct assault on one of the shortest routes to splitting Russian-held territory. Commercial satellite images have shown multiple lines of Russian defenses in the area, where Moscow’s forces have spent months laying mine belts, digging bunkers and setting out concrete barriers for tanks.

Pentagon officials and military analysts have warned against trying to make broad assessments about Ukraine’s success based on a few days of fighting in a few locations.

“This isn’t something you judge based on a few days of fighting,” Michael Kofman, the director of Russian studies at CNA, a research institute in Arlington, Va., said in a Twitter message on Friday. “The offensive will play out over weeks and likely months.”

Christiaan Triebert and Haley Willis contributed reporting.