Turkey on Thursday rejected US criticism over mass detentions of students during a month of escalating protests, telling Washington not to interfere in its internal affairs.

Hundreds of students and their supporters were detained this week after police violently cracked down on protests against President Recep Tayyip Erdogan's appointment last month of a loyalist as rector of Istanbul's elite Bogazici University.

Erdogan on Wednesday condemned the LGBT movement's role in the rallies and likened some student protesters to "terrorists".

The US State Department called Erdogan's "rhetoric" unacceptable, urging Turkey to embrace protests and "even speech that some may find uncomfortable".

The Turkish foreign ministry responded on Thursday by saying it would "advise those who dare to teach a lesson of democracy and law... to look in the mirror".

"No one should dare to interfere in Turkey's domestic affairs," it said in a statement.

But the statement responded only to Washington's criticism of the detentions, without referring to Erdogan's comments about the LGBT movement.

Erdogan on Wednesday said there was "no such thing" as LGBT, adding: "This country is... moral, and it will walk to the future with these values."

The State Department said it "strongly (condemned) the anti-LGBTQIA rhetoric surrounding the demonstrations". 

The diplomatic row threatens to undermine Erdogan's efforts to build up a rapport with the new US administration of President Joe Biden, after enjoying a personal friendship of Donald Trump.

Biden is expected to take a tougher line on Erdogan, pressing him on human rights.

Turkish interior ministry spokesman Ismail Catakli said on Thursday that 528 people had been detained in connection with the protests this week.

Two were remanded in custody and 498 were released, including 108 who were freed under judicial controls, Catakli said, while the fate of the other 28 remained unclear.

The Istanbul governor's office earlier said police were on the hunt for eight more suspects, of whom it said five had been caught.