Displaced Iraqis from the Yazidi community ride in the back of a truck across the Iraqi-Syrian border at the Fishkhabur crossing, in northern Iraq, on August 11, 2014. At least 20,000 civilians, most of whom are from the Yazidi community, who had been besieged by jihadists on a mountain in northern Iraq have safely escaped to Syria and been escorted by Kurdish forces back into Iraq, officials said. (Ahmad Al-Rubaye/AFP/Getty Images)

THE NEW U.S. military operation in Iraq appears to have been successful thus far in its limited aims. President Obama announced Thursday that the siege of a mountainous area where thousands of members of the Yazidi minority were stranded had been broken, and “we helped save many innocent lives.” U.S. airstrikes also appear to have turned back an Islamic State advance toward the capital of Iraqi Kurdistan, and Kurdish forces have received fresh supplies of arms.

The U.S. mission nevertheless remains both open-ended and lacking in clear strategic objectives. Mr. Obama said air strikes would continue to defend Irbil and Baghdad, where U.S. personnel are stationed; he also said additional operations not involving combat troops could be undertaken to protect vulnerable populations. But he gave no indication of a broader campaign to reduce the reach of the Islamic State, which now controls a New England-sized territory across Iraq and Syria, including the major city of Mosul, five oil fields and Iraq’s largest dam.

The limits Mr. Obama has placed on U.S. action make little sense in the context of this extremist entity and the interconnected conflicts across the region. Mr. Obama was right to rescue the tiny Yazidi minority, which was threatened with genocide, but why not also defend Syrians and Lebanese under threat of massacre by the Islamic State? It’s a vital U.S. interest to protect Kurdistan, a relatively stable and pro-Western enclave, but it’s also critical that moderate Syrian opposition forces under siege in Aleppo not be destroyed by the Islamic State or the regime of Syrian President Bashar al-Assad.

Mr. Obama still lacks an integrated policy for Iraq and Syria, though the Islamic State cannot be defeated unless it is attacked in both countries. By his own account, the president’s strategy for Iraq envisions a prolonged process of fashioning a new Iraqi government and rebuilding the Iraqi army before an offensive can begin against the extremists. It’s not at all clear that the Iraqi political and military restructuring the White House hopes for can be achieved; a decade of history suggests that at best there will be partial results. Meanwhile, senior U.S. intelligence and homeland security officials are sounding alarms about the danger posed by the thousands of Westerners recruited by the Islamic State; Attorney General Eric H. Holder Jr. said this situation was “more frightening that anything I think I’ve seen as attorney general.”

It seems clear that answering that threat should not be made contingent on a solution to Iraq’s sectarian divisions, which are part of a broader regional conflict. Rather, the United States should focus on weakening and eventually eliminating the toxic entities that are destroying the region and threatening vital U.S. interests: the Islamic State and the Assad regime. It can do so by directly supporting friendly forces willing to fight those enemies, including the Kurdish pesh merga, the Free Syrian Army, vetted Iraqi army units and Iraqi Sunni tribes opposed to al-Qaeda. U.S. backing should include arms delivery, training and advising, intelligence support and air power.

If the Islamic State and the Assad regime can be defeated or at least placed on the defensive, political solutions that address the Sunni-Shiite conflict in Iraq and Syria, Kurdish aspirations for self-determination and the protection of minority groups will come more easily. The idea that Iraqis will somehow solve these problems independently of Syria and with minimal U.S. support is a convenient but dangerous illusion.