ANKARA/ISTANBUL (Reuters) - Turkish President Tayyip Erdogan said on Monday he believed relations with Washington would improve under President-elect Donald Trump and that the two NATO allies would reach an easier consensus on regional issues.

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Ties between the United States and Turkey - which has the second largest army in the NATO alliance and is key to the U.S.-led fight against Islamic State in Syria and Iraq - have deteriorated sharply since a failed military coup in July.

Erdogan and the government blame the abortive putsch on Fethullah Gulen, an Islamic cleric who has lived in self-imposed exile in Pennsylvania since 1999, and want him extradited. Gulen denies involvement in the coup.

Ankara has also been angered by U.S. support for a Kurdish militia group fighting Islamic State in Syria. Turkey sees the group as an extension of the PKK, which has waged a three-decade insurgency for Kurdish autonomy in southeastern Turkey and has been behind a spate of recent bombings.

“I believe we will accelerate dialogue when Mr Trump takes office,” Erdogan told a conference of Turkish ambassadors. “I believe we will reach a consensus with Mr Trump, particularly on regional issues.”

Foreign Minister Mevlut Cavusoglu told the envoys he believed Trump would not make what he called the same mistakes as the outgoing U.S. administration. Washington should extradite Gulen and end cooperation with the Syrian Kurdish YPG militia.

CONSPIRACIES

The United States has long been cast by some nationalist and Islamist elements as the guiding hand behind plots to undermine Turkey. Conspiracy theories have reached fever pitch in recent weeks, and in some cases been fuelled by elected officials.

The Islamist Yeni Akit newspaper said last week that a New Year’s Day shooting which killed 39 people in an Istanbul nightclub, claimed by Islamic State, was a “joint production” by the CIA, Israel’s Mossad spy agency, and what the Turkish government calls “FETO”, the Gulenist Terror Organisation.

“Daesh (Islamic State), the PKK and FETO are terror groups with strings linked to the same centre. Along with many secret services, the most active user is the CIA,” Samil Tayyar, a ruling AK Party lawmaker, said on Twitter after the attack.

Such comments prompted a rebuke last Thursday from U.S. State Department spokesman John Kirby: “Incendiary and false accusations about the U.S. in Turkish media are offensive and could endanger our citizens. Must stop,” he wrote on Twitter.

Any honeymoon with the new Trump administration is likely to be short-lived.

U.S. officials have made clear the issue of Gulen’s extradition is a matter for the courts, not a question of political will, and that such a process could take years.

Trump’s policy on the Kurdish YPG is unclear. He has indicated, though, that his priority in the multi-sided Syrian conflict will be fighting Islamic State - not forcing President Bashar al-Assad to leave power as Turkey has long desired.