Locals use an isolated building to watch an Islamist State attack on Kobane city in Syria, near Turkish-Syria border Suruc district, near Sanliurfa, Turkey, 04 October 2014. (Sedat Suna/EPA)

TWO MONTHS after the United States began airstrikes in Iraq, and two weeks after they were extended into Syria, the forces of the Islamic State are still advancing. Last week they captured the Iraqi towns of Hit and Kubaisa, northwest of Baghdad. On Tuesday they appeared close to overrunning Kobane, a strategic city on the border between Syria and Turkey that is populated by Kurds. The enemy victories are happening in spite of U.S. and allied airstrikes and resistance from local forces. They suggest that the U.S. air campaign is failing to achieve the minimal aim of stopping the expansion of the Islamic State — much less “degrading” and “destroying” it.

Why can’t the U.S.-led coalition prevent a ragtag insurgent army from overrunning large towns? The answers speak to the limitations imposed on the military campaign by President Obama as well as the continuing political complications of fighting the Islamic State. Military analysts point out that U.S. strikes on Islamic State forces around Kobane have come late and in small handfuls — not enough, as of Tuesday, to turn back thousands of fighters armed with tanks and artillery. In contrast with the successful 2002 air campaign against the Taliban in Afghanistan, U.S. pilots cannot rely on Special Forces spotters to identify targets. Mr. Obama has ruled out such ground personnel despite requests from military commanders.

Kobane has also been a victim of the ambivalent approach to the war of the Turkish government of Recep Tayyip Erdogan, which says it supports the fight against the Islamic State but has so far refused to join it or even to allow U.S. planes based in Turkey to carry out missions in Syria. Erdogan’s government is reluctant to help the Kurdish militia in Kobane because it is allied with a Kurdish rebel group in Turkey, even though Turkey has a truce with its Kurds. Turkish tanks are positioned on a ridge overlooking Kobane but have not joined the battle; instead, Turkish troops have been trying to stop Kurdish reinforcements from crossing into Syria.

On Tuesday Mr. Erdogan conceded that “Kobane is about to fall” and offered it as evidence that “the problem of the [Islamic State] . . . cannot be solved via air bombardment.” He is still pressing the United States to establish a no-fly zone in Syria as well as a protected area for Syrian rebels. But those measures would be aimed less at the Islamic State than at the Syrian regime of Bashar al-Assad — and Mr. Obama still refuses to take steps directly aimed at defeating the Assad regime.

Contradictions such as these are allowing the Islamic State not only to survive but also to expand. Independent military analysts believe that, in addition to targeting Kobane and towns in western Iraq, the Islamic State is readying a full-scale assault on Baghdad. To turn back the offensive, U.S. forces will need to step up the tempo of operations and react more quickly to movements on the ground, which will be difficult without ground personnel. Turkey, with its powerful army, must be brought into the fight, and that will require U.S.-Turkish agreement on a strategy for dealing with the Assad regime.

For now, the U.S. operation in Iraq and Syria is defined mainly by its limitations. The restrictions Mr. Obama has imposed on his commanders are not compatible with the objectives he has asked them to achieve.