WASHINGTON — Images of angry mobs in Arab cities burning American flags and attacking U.S. diplomatic posts suggest the Muslim world is no less enraged at the United States than when President George W. Bush had to duck shoes hurled at him in Baghdad.

But more than three years after President Barack Obama declared in Cairo that he would seek “a new beginning” in U.S.-Muslim relations, a closer look reveals strides as well as setbacks.

One U.S.-led war is over and another is receding, although there are questions about whether America has made lasting gains in Afghanistan. The Arab Spring revolution, a spontaneous combustion that happened independent of Western influence, has given people new power and hope as well as democratic elections the U.S. supports.

But peace between Israel and the Palestinians is nowhere in sight, Iran is seen as a menace and broad mistrust with America is still deep and explosive across much of the Muslim world.

As nations across North Africa and the Middle East move chaotically toward democracy, they and Washington have settled into a wary, redefined relationship. Obama is not ready to call Mohammad Morsi, the popularly elected Egyptian president, an ally, and the democratically elected Iraqi president, Nouri al-Maliki, has dismissed U.S. demands that he stop Iran from using Iraqi airspace to fly weapons to Syria for use against anti-government rebels.

Such is the complicated progress report that Obama carries toward the United Nations General Assembly next week, his final moment on a world stage before the U.S. election on Nov. 6. For that election, Pew Research Center polling shows Obama has a clear edge over Republican Mitt Romney in handling foreign policy in general and problems in the Middle East specifically.

Across the world his standing remains markedly lower in predominantly Muslim nations. However, Leila Hilal, a Mideast expert at the New America Foundation, said Obama may have made more progress toward improving relations than critics say.

“Obama inherited a very damaged U.S. credibility in the region,” she said, and so it would be unrealistic to think that his “new beginning” would take hold fast.

“There’s only so much one president can do, given the history” of perceived insults by the U.S., she said. Those range all the way from the American invasion of Iraq to, more recently, the privately made anti-Islam video that ridicules the prophet Muhammad and triggered major protests across the Muslim world.

The question of the Obama administration’s relationship with that Muslim world came under new election-year scrutiny when four Americans, including Ambassador Chris Stevens, were killed in a Sept. 11 attack on the U.S. Consulate in Benghazi, Libya.