“It may be slow when you look at the numbers,” Ms. Malyar said, “but the progress is confident.”

As Ukraine hunts for vulnerabilities in Russian lines that have been fortified over months, independent analysts say that Kyiv’s advances in both the east and the south in recent days are better measured in yards than miles.

Ukraine claimed to retake a string of villages around a river in the Zaporizhzhia region over the weekend and on Monday. But the main Russian defensive lines still lie 15 to 20 kilometers (nine to 12 miles) beyond current Ukrainian positions, according to Jack Watling, a research fellow and specialist in land warfare at the Royal United Services Institute in Britain.

Ukrainian troops must get past multiple layers of Russian fortifications before that, Mr. Watling wrote in a paper published on Wednesday. The slow progress Kyiv is reporting in the early days of the counteroffensive reflects bloody fighting to clear fox holes and hand-dug trenches along Russia’s frontline fighting positions, he said.

“Behind these were complex minefields of anti-tank and antipersonnel mines,” he wrote. Ukrainian forces trying to push forward into these areas will most likely be closely tracked by Russian drones and targeted by Russian artillery. And near the main Russian defensive lines are “properly dug trenches and concrete-reinforced firing posts, tank obstacles, ground-laid cable to coordinate artillery strikes and even more mines,” he wrote.

While Ukraine has yet to deploy the bulk of its offensive force, its initial push, he said, is aimed at getting Russia to bring reserve troops from far behind the front lines to shore up areas under pressure.

“Once these troops are pulled forwards, it will become easier to identify the weak points in the Russian lines,” Mr. Watling wrote.

But as Ukrainian forces advance, Mr. Watling and other analysts say, they will also be covered by fewer air defenses and could come under even more sustained attack by Russian helicopters and warplanes. On Tuesday, Russian officials said attack helicopters had struck a Ukrainian position near one of the newly recaptured villages.

Ukraine has already lost several Western-supplied tanks and armored vehicles in the initial days of its offensive, according to videos and photographs verified by The New York Times, and analysts say its forces may also have suffered significant casualties, although Kyiv does not disclose those losses.

On the Russian side, too, there are signs that the fighting has been bloody: Dozens of dead Russian soldiers and burned-out Russian vehicles littered the road to Storozhove in the Donetsk region, one of the villages retaken by Ukraine, according to journalists from Sky News who visited the area on Wednesday.

But Ms. Malyar on Thursday highlighted how Western-supplied equipment was helping to save the lives of Ukrainian soldiers.

She showed a photograph of an American-made Bradley fighting vehicle that came under a hail of Russian rocket fire and sustained a direct hit. She then showed a photograph of the crew that staffed the vehicle. They all escaped and survived, she said.

The Bradley vehicle, she said, “helps save the most precious thing — the lives of military personnel. And steel can always be restored.”