DRUZHKIVKA, Ukraine — The last bridge to the last city standing between the Russian army and control of easternmost Ukraine has collapsed, as soldiers engaged in pitched street battles for what little is left of the city and scores of civilians remained stranded under unrelenting bombardment.
Even after the final remaining bridge between the city, Sievierodonetsk, and Ukrainian-held territory to the west was toppled, Ukrainian officials insisted that supplies could still reach their soldiers fighting in the ruined city. But evacuating civilians and wounded troops may become far more challenging, and it appeared increasingly unlikely that the city’s outgunned defenders could hold out for long.
Almost four months after the forces of Russia’s president, Vladimir V. Putin, attacked, Ukraine has been largely reduced to harrying the better-equipped invader, making each patch of ground as bloody for it to win as possible, but failing in recent weeks to secure any decisive victories, and losing many of its own soldiers and citizens in the process.
Ukraine’s president, Volodymyr Zelensky, insists that his country may yet prevail if it is given more powerful weaponry, but as Western military leaders prepared to meet in Brussels on Wednesday and Thursday, some officials sounded dubious, with talk again turning to what an end to the war might look like — and how to bring it about.
Still, American officials appeared at pains on Tuesday not to overstep.
“We’re not going to tell the Ukrainians how to negotiate, what to negotiate and when to negotiate,” Colin H. Kahl, the under secretary of defense for policy, said at a security conference in Washington. “They’re going to set those terms for themselves.”
The fall of Sievierodonetsk would put Mr. Putin an important step closer to seizing Ukraine’s industrial heartland, the eastern Donbas region, where he has directed Russia’s assault after failing early in the war to take the largest Ukrainian cities, including Kyiv, the capital, and Kharkiv.
In one Donbas province, Luhansk, Russian troops and their separatist allies now hold all but a small pocket containing Sievierodonetsk and, across the Siversky Donets River, the city of Lysychansk. And they control more than half of a neighboring province, Donetsk.
It was unclear whether it was Russians or Ukrainians who, late Monday or early Tuesday, toppled the last bridge across the river between the two cities, which was already barely usable. It was also unclear which side might pay a higher price for its loss.
Destruction of the bridge raises the risk to Ukrainian forces of being trapped in Sievierodonetsk. But it could also complicate the Russian advance, making it more difficult to mount a frontal assault on Lysyschansk, which lies on higher ground — an attack would require the Russians to expose their troops while crossing the river below.
Better Understand the Russia-Ukraine War
Mr. Zelensky’s government has asked the West for a vast increase in sophisticated weapons and tools, like rocket launchers, howitzers, drones and tanks. The increasingly desperate pleas appeared likely to put added pressure on 40 Western allies when they meet in Brussels along with Jens Stoltenberg, NATO’s secretary general, and the American secretary of defense, Lloyd J. Austin III.
Some Western officials say Mr. Zelensky may not have a viable strategy to win the war. The Ukrainians have had some success fighting at relatively close ranges, and the Russians have countered by relying on their immense advantage in longer-range artillery and missiles, pounding cities and towns to rubble before sending in troops.
But a war of attrition — Ukraine has been losing as many as 200 soldiers a day in the fighting — favors Russia for the simple reason that it has more soldiers to lose. And, Mr. Zelensky has said, Mr. Putin has shown he is willing to treat his own troops as “cannon fodder.”
But for now, at least publicly, the United States is endorsing the Ukrainian desire to keep up the battle.
“Our role,” said Mr. Kahl, the defense under secretary, “is to help them make sure that they can defend themselves against the Russian onslaught — and they’ve been doing an unbelievably courageous job at that — and to strengthen their hand whenever the negotiations do happen.”
The profession of support notwithstanding, the remarkable initial unity in response to Russia’s invasion seems to be fraying among the Western allies who have shipped weapons to Ukraine and imposed a broad array of financial sanctions on Russia.
Some leaders in Central and Eastern Europe, with its long experience of Soviet domination, have strong views about the need to teach Russia a painful and lasting lesson — with some going as far as to reject the idea of even speaking to Mr. Putin.
But France, Italy and Germany, among the continent’s biggest and richest countries, are anxious about the prospect of a prolonged, stalemated war that raises the risk of drawing NATO into the fighting and would further damage their economies as they grapple with rising inflation and fuel prices.
The United States has sought to balance two oft-conflicting goals. The first is for Ukraine to emerge as a vibrant, democratic state — exactly what Mr. Putin is seeking to crush. The second is President Biden’s goal of avoiding direct conflict with Russia, which he has repeatedly called World War III.
The European Union is seriously considering starting Ukraine along a path to membership, despite its history of corruption and political instability, which diplomats say might help entice Ukraine into agreeing to a negotiated end to the fighting. It is not clear, though, what might persuade Mr. Putin to make a deal, particularly as long as his forces keep taking more cities and towns.
But not Sievierodonetsk, not just yet, its mayor, Oleksandr Stryuk, said on national television. “Russian troops are trying to storm the city” he said, “but the military is holding firm.”
Fighting in Sievierodonetsk, an industrial city, has been fierce. Videos posted online have shown gun battles on ravaged streets. They are reminiscent of the fighting in Mariupol in the country’s south, before that city fell to the Russians, where blasted-out apartment buildings were turned into machine gun nests and every window could hide an enemy sniper.
The close nature of the fighting largely negates — for the moment — Russia’s superiority in artillery barrages and airstrikes. Only dozens of yards have separated Russian and Ukrainian forces at times, meaning that friendly fire is a common concern.
Urban fighting favors the defender, but it is not sustainable for long as casualties mount and ammunition runs dry. The loss of the bridge may prove decisive on that second score.
“Our guys did not leave Sievierodonetsk; they are still fighting for every meter of our Motherland,” Oleksandr Voronenko, a military police officer stationed near the city, said Tuesday evening.
With Russian firepower concentrated in the east, Ukraine claimed some successes elsewhere in the country on Tuesday. The Ukrainian military said it was outmaneuvering Russian forces in the south, steadily reclaiming towns and villages and bringing its forces to within 12 miles of the Russian-occupied city of Kherson.
Given its lack of heavy artillery, it is unlikely that Ukraine will launch a broad offensive any time soon to try to take back Kherson. But Russian forces are also having to contend with a growing insurgency made up of civilians and former soldiers.
“The enemy continues to fight, but our units are gradually forcing him to liberate positions and test the strength of the second and third lines of defense, and in some cases to retreat further,” the Ukrainian southern military command said in a statement.
While both sides have suffered heavy losses and neither seems poised to deliver anything resembling a decisive blow, the fighting remains intense in many places. The entire front line dividing Ukrainians and Russians stretches over a twisting 1,500 miles, with active fighting reported along more than 680 miles, the commander of the Ukrainian armed forces said this week.
Thomas Gibbons-Neff reported from Druzhkivka, and Eric Schmitt from Washington. Reporting was contributed by Eric Nagourney from New York, Natalia Yermak from Druzhkivka, and Andrew E. Kramer andValerie Hopkins from Kyiv.
