It would be nice to think that Rick Perry’s withdrawal from the GOP presidential race last week had something to do with the boneheaded remark he made about Turkey recently in the Myrtle Beach, S.C., debate. The exchange with Bret Baier of Fox News easily qualifies as the foreign policy low point of the presidential campaign so far — which is saying something. It also points to a problem in American — and in particular, Republican — understanding of the changing Middle East that is much bigger than Perry.

Baier delivered a mostly accurate but extremely one-sided description of the government of Recep Tayyip Erdogan, saying that since his “Islamist-oriented party took over . . . the murder rate of women has increased 1,400 percent. Press freedom has declined to the level of Russia. [Erdogan] has embraced Hamas, and Turkey has threatened military force against both Israel and Cyprus.” Then he asked: “Do you believe Turkey still belongs in NATO?”

Perry responded: “Well, obviously when you have a country that is being ruled by what many would perceive to be Islamic terrorists . . .

Islamic terrorists? This, mind you, is about a government that has just stationed an advanced radar on its territory that could be used to track and shoot down missiles from Iran; that joined the NATO operation against Moammar Gaddafi in Libya; that has become the host of the opposition to Syrian dictator Bashar al-Assad; and that, having repeatedly won free democratic elections, amended Turkey’s constitution to expand rights for women, ethnic minorities and unions.

Okay — that, too, was a one-sided account of the Erdogan record. But that is precisely the point: Turkey has become a complex, dynamic, difficult, sometimes infuriating, sometimes very helpful and indisputably important ally of the United States. In that sense, Erdogan’s government is a paradigm of the relationships U.S. administrations will be managing — if we are fortunate — in Egypt, Iraq and elsewhere in the Arab Middle East during the coming decade.

The reality is that, like it or not, “Islamist-oriented” governments are about to become the new normal in a region dominated for decades by secular autocrats and pro-American generals. So the crude bias about Muslim movements that is baked into the worldview of many U.S. conservatives — that they are inevitably fundamentalist, anti-democratic, anti-Israel and anti-American, if not explicitly “terrorist” — has become a serious liability. If heeded, it will make it impossible for this administration and future ones to navigate the region’s new politics and preserve crucial alliances.

Some Islamic movements may turn out like Hamas and Hezbollah — implacably hostile. But others, like Egypt’s Muslim Brotherhood, are likely to weave through an ambiguous middle ground, trying to balance the need for Western investment and the secular aspirations of their populations with their religious ideology. The right way to respond to them is to be nimble: tolerate some turbulence, roll with some punches, push back against others and keep pressing leaders to stick to democratic principles.

This is pretty much how Barack Obama has handled Erdogan and his government — with a net result that, for now at least, looks positive. I’ve written recently about Obama’s failures on several of his signature foreign policy initiatives. But his handling of Turkey, and its mercurial leader, may stand as one of his best accomplishments.

Fortuitously, Obama began courting Erdogan from the beginning of his administration, making Istanbul the site of one of his first foreign trips and delivering a speech in which he pledged to build stronger relations between the United States and Turkey as well as the Muslim world more generally.

Some big disappointments followed. Erdogan turned on Israel even before Israeli commandos’ disastrous interception of a Turkish ferry trying to break the Gaza blockade in 2010. He tried to undercut, and then voted against, Obama’s attempt to win U.N. Security Council approval for sanctions against Iran. At home, his attacks on critical journalists and the jailing of hundreds of military officers and secular activists on dubious charges of coup plotting prompted public criticism from the U.S. ambassador and eventually, Secretary of State Hillary Rodham Clinton.

But Obama kept working on the Turkish leader, calling him more often than any foreign partner other than Britain’s David Cameron. A relatively close personal rapport has been the result. Asked about his foreign relationships in an interview last week, Obama named Erdogan among five world leaders with whom he said he had forged “bonds of trust.”

Administration officials say they see a convergence of U.S. and Turkish policies in the past year — on Libya, Syria, Iran and the Arab Spring more generally. While Erdogan’s drift toward domestic autocracy remains a major concern, some officials believe a new constitution his party is drafting will lead to better checks and balances, and fewer journalists in prison.

This will not make Turkey an ideal ally, or remedy its still-troubled relations with Israel. But it’s much better than turning its Islamists into adversaries — or failing to distinguish them from terrorists.