As Russia relies on overwhelming destructive power to advance a mile or two a day in eastern Ukraine, Ukrainian soldiers fighting some 400 miles to the south are working steadily to drive out Russian front-line positions across a stretch of steppe and swamp. The fighting is fierce on both fronts and how the two campaigns play out is critical to understanding the state of the war, as concern grows that a protracted conflict will impose new economic costs on Ukraine’s allies. President Vladimir V. Putin of Russia said last week that he believed he would wait for the West to pull out. While the Russian leader rarely acknowledges Russian losses or defeats, military analysts said the blows his army has taken raise questions about whether it can withstand broad offensive operations after the end of its campaign to seize Luhansk province. Russia has committed the bulk of its combat forces to the capture of Lysychansk, the last urban center in Luhansk still controlled by the Ukrainian government, and it could fall any day. Russia has sent thousands of additional troops to the east in recent weeks to bolster its offensive in neighboring Donetsk province, where it will likely try again to crush heavily fortified Ukrainian positions with its large arsenal of artillery, missiles and air force, even if ground forces are reduced. How reduced each army is after more than four months of war is an open question. Kyiv publishes only general estimates of its losses, and Moscow says almost nothing. British Defense Chief Ben Wallace said last week that 25,000 Russian soldiers had been killed since the start of the war. The figure, which could not be independently confirmed, is the highest estimate provided by a senior Western official. The Ukrainian government has acknowledged that it has suffered staggering casualties, with hundreds of casualties every day. Even if Russia can push deeper into Donetsk, its military has struggled to maintain an advance along multiple lines of attack in different parts of a country roughly the size of Texas. The Russian defeat on Thursday at Snake Island in the Black Sea, where their troops were forced to retreat under a sustained Ukrainian bombardment, underlined how dependent the Russians are on their superiority in heavy weapons. The Russian withdrawal from the island was expected to undermine Moscow’s control over vital grain routes from Odessa. And when Russian missiles hit a residential building and a recreation center near Odessa, killing at least 21 people on Friday, Ukrainians saw it as an act of revenge. “This was an act of revenge for the successful liberation of Snake Island,” Yevhen Yenin, the first deputy minister of internal affairs, said in an interview. He scoffed at Russian claims that the withdrawal from the island was a gesture of “good will”. A satellite image from Maxar Technologies showed smoke rising from Snake Island on Thursday. Credit…Maxar Technologies/Via Reuters With its forces stretched thin, Russia has been trying for months to fortify its defensive positions in the south, where Ukraine has recaptured parts of the Kherson region west of the Dnieper river that Russia seized early in the war. The Ukrainian military said the Russians had been driven from perimeter defense positions at several locations and that Ukrainian troops were operating within 20 miles of the city of Kherson. Senior US Defense Department officials said last week that the Ukrainians were not only retaking villages, but also showing the ability to hold retaken territory. But military analysts have warned that despite Ukraine’s gains in the south, they are unlikely to be able to launch a broad offensive and advance soon on the city of Kherson, the only provincial capital to fall to the Russians. At the moment, Ukrainian forces are fighting back in the north and south of the city. At the same time, rebels inside Kherson have stepped up a campaign to assassinate Russian proxy leaders and aid the Ukrainian military by engaging in sabotage operations and assisting with direct fire on Russian targets. On Thursday, Ukraine’s military’s southern command said its forces launched rocket and artillery attacks on 150 targets, killing more than 40 Russian soldiers and destroying scores of Russian artillery and armored personnel carriers. The claims could not be independently confirmed, but data from NASA fire-monitoring satellites showed activity across the southern front. —Marc Santora and Roger Cohen