WASHINGTON—Five months ago, President Donald Trump hailed Turkish President Recep Tayyip Erdogan as a friend and said the two North Atlantic Treaty Organization allies had never been closer.

But Mr. Trump’s optimistic view masked a more complicated reality: Turkey quickly has become one of the Trump administration’s most vexing relationships. Relations between the two countries, Secretary of State Rex Tillersonconceded last week, are at a “crisis point.” And there appears to be no clear road map for the two countries to overcome...