The United States said it remained concerned about Turkey's censorship of free speech, and women's groups protested in Istanbul on Saturday, after the arrest of pop star Gulsen over a past quip she made about religious schools.

The singer-songwriter was jailed on Thursday pending trial on a charge of incitement to hatred after a video of her on-stage remark in April was broadcast by a pro-government media outlet.

While several state ministers condemned Gulsen's words, her arrest drew a fierce response from critics who see President Tayyip Erdogan's government as bent on punishing those who oppose its conservative views.

A US State Department spokesperson said it remains concerned about widespread efforts in Turkey to restrict expression via censorship and judicial harassment following Gulsen's detention.

Protesters in Istanbul criticised what they called inconsistency between the judiciary's inaction towards violence against women and the artist's speedy investigation and arrest. Many say Gulsen was targeted for her liberal views and support for LGBT+ rights.

"Hundreds of women would be alive today if men who assaulted other women were captured as fast as Gulsen was," organisers of the Istanbul protest told demonstrators through a loudspeaker.

Her arrest is the latest injustice against "women who don't fit the mould," or are not "the type of woman the government wants," they said.

In the video of her performance in April, Gulsen refers to a musician in her band and says in a light-hearted manner: "He studied at an Imam Hatip (school) previously. That's where his perversion comes from."

Erdogan, whose Islamist-rooted party first came to power two decades ago, himself studied at one of Turkey's first Imam Hatip schools, which were founded by the state to educate young men to be imams and preachers but have since exploded in number.

Gulsen on Thursday apologised to anyone offended by her remarks, saying they were seized upon by some who want to polarise society.