After months of anticipation, Ukraine’s forces — newly trained on complex warfare tactics and armed with billions of dollars in sophisticated Western weaponry — launched operations on multiple fronts in the past week in an effort to dislodge entrenched Russian military units, a counteroffensive that many officials in the United States and Europe say could be a turning point in the 15-month war.

Much rides on the outcome. There is little doubt the new military drive will influence discussions of future support for Ukraine as well as debates about how to guarantee its future. What remains unclear, though, is exactly what the United States, Europe and Ukraine view as a “successful” counteroffensive.

Publicly, American and European officials are leaving any definition of success to President Volodymyr Zelensky of Ukraine. For now, Mr. Zelensky has not laid out any public goals, beyond his oft-stated demand that Russian troops must leave the whole of Ukraine. He is known as a master communicator; any perception that he is backing off that broad ambition would risk undermining his support at a critical moment.

Privately, U.S. and European officials concede that pushing all of Russia’s forces out of occupied Ukrainian land is highly unlikely. Still, two themes emerge as clear ideas of “success”: that the Ukrainian army retake and hold on to key swaths of territory previously occupied by the Russians, and that Kyiv deal the Russian military a debilitating blow that forces the Kremlin to question the future of its military options in Ukraine.

Some battlefield success, whether by decimating Russia’s army, claiming some territory or both, could help Kyiv secure additional military aid from Europe and the United States. It would also build confidence in allied capitals that their strategy of remaking Ukraine’s forces into a Western-style military is working. And most importantly, such an outcome would build more support in Europe for some sort of long-term security guarantee for Kyiv and strengthen Ukraine’s hand at a bargaining table.

Success is not guaranteed. Throughout the war, the Ukrainian army, with deeply motivated troops, creative military operations and advanced Western weaponry, has outperformed Russia’smilitary. But the Ukrainians have also found it difficult to dislodge the Russians from their entrenched defensive positions in the last few months, with the front lines barely moving.

Nevertheless, Ukraine has shown it can launch successful offensives, like the one last year in which it took a large amount of territory east of Kharkiv and, after a long fight, regained the southern city of Kherson.

American intelligence agencies have assessed that the most likely scenarios are smaller Ukrainian victories in the opening stage of the fighting, like retaking some parts of the Donbas or pushing Russia out of agricultural and mining areas in southeastern Ukraine.

Seizing the nuclear plant in Zaporizhzhia would be both a symbolic and strategic victory, putting one of the world’s largest nuclear plants and an important source of electricity back in the hands of Ukraine.

American and European officials say it is key for Ukraine to cut off, or at least squeeze, the so-called land bridge: the large swath of territory Russia seized between its border and the peninsula of Crimea, which has become a main supply route for the military stronghold it has built there.

Kyiv wants to reclaim its southeastern coast on the Sea of Azov. If Ukraine can drive its forces to the coastline, cutting off Crimea, Mr. Zelensky could count that as a huge win. But even if Ukrainian forces did not reach the sea, and instead took midsize cities in southern Ukraine, that would effectively narrow the land bridge.

From those positions, Ukrainian forces could use medium-range artillery to threaten Russian command posts on Crimea and any military supply convoys Russia sends along the coastline. While the Russian forces in Crimea are currently well supplied, American officials said, laying siege to the land bridge would make the winter difficult for them.

Retaking land is one thing, but what is crucial, American officials said, is for Ukrainian forces to hold on to it.

Essentially, the United States and its allies will be looking at the counteroffensive for evidence that their plan of remaking the Ukrainian army into a modern force that fights with NATO tactics, and that can use complex maneuvers and advanced equipment to allow a smaller force to defeat a larger one, is sound.

A strong showing by Ukraine will have the added benefit of further eroding Russian troops’ morale. Right now, the Russian army faces critical shortages of weapons and personnel — Moscow was forced to take decades-old tanks out of storage to use in fighting and has relied on barely trained conscripts. Those shortages should prevent Russia’s forces from capitalizing on any Ukrainian missteps or mounting their own offensive in the coming months.

“Moscow has suffered military losses that will require years of rebuilding and leave it less capable of posing a conventional military threat to Europe and operating assertively in Eurasia and on the global stage,” Avril D. Haines, the director of national intelligence, told the Senate last month.

Still, Russian forces are beginning to get better — they are improving their tactics and practicing better defensive operations. War always favors the defenders, something the entrenched Russians may be able to use to their advantage during Ukraine’s counterattack.

For now, the Russian air force has been largely absent in the war, with Ukrainian air defense batteries threatening Russian bombers and fighter planes. The United States and its allies have tried to shore up the shortages in Ukraine’s air defense equipment. But if Russia flies more aggressive bombing runs into Ukraine, it could pose a challenge during the counteroffensive.

American and European officials say a vital goal of the counteroffensive should be to weaken the Russian military further. Russian forces have taken huge numbers of casualties in fighting this year in Bakhmut in eastern Ukraine. Success, as one NATO ambassador put it, would be pushing Russia back and killing a lot of Russian troops.

Another potential scenario, according to U.S. intelligence agencies, is that the Russians make an error, for example by putting their troops in the wrong place or defending a trench line too lightly, which could allow Ukraine to punch through the lines and execute a devastating blow to Russian troops.

Of course, some allied officials worry that Ukraine might be too successful. A huge loss of soldiers could force Mr. Putin to mobilize a broader swath of his population to build up his army.

And while U.S. officials have said the risk of Mr. Putin’s using a nuclear weapon have receded, American intelligence agencies say total defeat in Ukraine or a loss of Crimea are two scenarios under which Mr. Putin could potentially order the use of a nuclear weapon.

A failed counteroffensive is easier to measure. If battle lines stay relatively unchanged, or Ukraine is unable to recapture a significant city, some officials in allied capitals or Congress will likely raise doubts about the war, especially if Ukrainians lose too many troops and a lot of equipment is destroyed.

The United States, NATO allies and Ukraine have been training about 30,000 troops on combined arms maneuvers — a complex style of warfighting that involves constant communication between tanks, artillery, fighter jets and infantry forces — for the express purpose of leading the counteroffensive.

If the Ukrainians fail to make significant gains using these maneuvers, that could put in doubt the long-term U.S. strategy of strengthening Ukraine by giving them even more sophisticated weaponry and complex training.

In essence, according to European diplomats, failure would look like a Ukrainian army that has not learned to fight, has lost the equipment given to them in recent months and gained no territory to show for that — with a Russian military poised to renew its drive.

Despite some early casualties, and tough Russian defenses in the east, American officials are optimistic that Ukraine will make enough gains, however incremental, to call the fighting a success.

Both Ukraine and Western allies have invested in the counteroffensive because, no matter the precise result, it will set the stage for the next phase of the war. The American and British plan to help secure Ukraine involves building support for robust security guarantees from the United States and NATO countries as well as pushing a plan to build closer economic ties between Kyiv and European countries.

Crucially, if the counteroffensive leaves Russia weakened, it could be forced to engage in meaningful dialogue with a stronger Ukraine.

Biden administration officials are careful to say their support for Ukraine will not hinge on the success of the counteroffensive.

Speaking with Rishi Sunak, the prime minister of Britain on Thursday, President Biden brushed aside questions of future funding for Ukraine’s fight.

“I believe we’ll have the funding necessary to support Ukraine as long as it takes,” Mr. Biden said.

But realistically, success or failure could have an impact on support inside a fractious U.S. Congress, which has to authorize any additional funding for Ukraine, as well as in Europe, where there are similar concerns about how long the war will last, how much it will cost and what it will do to the prices of energy and food in the longer run.

Whatever the outcome of the counteroffensive, American and European officials agree that for now, Mr. Putin is in no mood to negotiate. But Mr. Putin understands raw power, and that is what makes the counteroffensive so important. If it is followed by continued Western support and security guarantees, that at least has the possibility of changing the calculus in Moscow.

David E. Sanger contributed reporting.

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