Speaking last week at a closed-door presentation hosted by Morgan Stanley, John R. Bolton, the former national security adviser, said Mr. Trump often confuses personal relationships with national relationships when it comes to setting policy. He cited as an example the president’s reluctance to confront Mr. Erdogan by imposing sanctions on Turkey over the Russian weapons purchase, a person who was in the room for his presentation said on Tuesday after NBC News reported a version of Mr. Bolton’s remarks.

On the Russian missiles, banking sanctions and other matters, Mr. Erdogan has deployed both his own son-in-law and Mr. Trump’s Turkish business partner, Mehmet Ali Yalcindag, as emissaries to the administration, sometimes through Mr. Kushner, according to Turkish officials and public records.

In April, for example, Mr. Albayrak had come to Washington for a conference organized by Mr. Yalcindag at the Trump International Hotel. And in the middle of the event, Mr. Kushner summoned Mr. Albayrak to an impromptu meeting in the Oval Office, where Mr. Albayrak successfully pressed Mr. Trump to hold back the sanctions against Turkey for buying Russian weapons.

Both leaders appear to favor family or business connections as back channels, several advisers to Mr. Erdogan said, in part because both share a suspicion that the agencies of their own governments may be conspiring against them.

The term “deep state,” in fact, first emerged in Turkey decades ago, long before it came into vogue among Trump supporters, and Mr. Erdogan’s advisers say he has cultivated Mr. Trump by emphasizing their shared struggles against such entrenched forces within their governments.

“The U.S. has an established order that we can call a deep state — of course they are obstructing,” Mr. Erdogan said this spring, explaining his hopes for the “bridge” between sons-in-law . “These obstructions are one of our main troubles.”