Four people have been killed in Iran's Kurdish region when security forces opened fire during protests over the death of a woman in police custody, a Kurdish rights group said, on a third day of turmoil over an event that has ignited nationwide anger.

Fresh protests broke out in the country over the death Mahsa Amini, a 22-year-old from Iran's Kurdistan province who was arrested by the "morality police" that enforces a strict dress code.

Ms Amini fell into a coma and died following her arrest in Tehran last week, sparking demonstrations in numerous areas including the capital.

Two of the people were killed as security forces opened fire on protesters in the Kurdish city of Saqez, Ms Amini's hometown, the Hengaw Human Rights Organisation said on Twitter.

It said two more were killed in the town of Divandarreh "by direct fire" from security forces.

There was no official confirmation of the deaths.

Demonstrations were held in the capital Tehran, including several of its universities, as well in Iran's second city, Mashhad, according to the Fars and Tasnim news agencies.

Protesters marched down Hijab Street, or "headscarf street", in the centre of Tehran denouncing the actions of the morality police, the ISNA news agency reported.

"Several hundred people chanted slogans against the authorities, some of them took off their hijab", Fars said, adding that "police arrested several people and dispersed the crowd using batons and tear gas".

A brief video released by Fars shows a crowd of several dozen people, including women who have removed their headscarves, shouting "Death to the Islamic republic!"

A "similar gathering" took place in the northeastern city Mashhad, the Tasnim agency reported.

The rallies today came a day after police made arrests and fired tear gas in the dead woman's home province of Kurdistan, where some 500 people had protested, some smashing car windows and torching rubbish bins, reports said.

The police have said Ms Amini fell ill as she waited with other women being held by the morality police, who enforce strict rules in the Islamic republic requiring women to cover their hair and wear loose fitting clothes in public.

But her father has repeatedly said his daughter had no health problems, adding that she had suffered bruises to her legs. He held the police responsible for her death.

He also criticised the "slow response" of the emergency services, adding: "I believe Mahsa was transferred to the hospital late."

Such police units enforce a dress code in the Islamic republic that demands women wear headscarves in public.

It also bans tight trousers, ripped jeans, clothes that expose the knees and brightly coloured outfits.

Police have insisted there was "no physical contact" between officers and the victim.

Tehran police chief General Hossein Rahimi said the woman had violated the dress code, and that his colleagues had asked her relatives to bring her "decent clothes".

He again rejected "unjust accusations against the police" and said "the evidence shows that there was no negligence or inappropriate behaviour on the part of the police".

"This is an unfortunate incident and we wish never to see such incidents again."

Students rallied, however, including at Tehran and Shahid Beheshti universities, demanding "clarification" on how Ms Amini died, according to Fars and Tasnim news agencies.

Her death has reignited calls to rein in morality police actions against women suspected of violating the dress code, in effect since the 1979 Islamic revolution.

Filmmakers, artists, athletes and political and religious figures have taken to social media to express their anger over the death, both inside and outside the country.

President Ebrahim Raisi, an ultra-conservative former judiciary chief who came to power last year, has ordered an inquiry into Ms Amini's death.

State television on Friday broadcast a short surveillance video that showed a woman identified as Ms Amini collapsing in the police station after an argument with a policewoman.

Amjad Amini, the victim's father, told Fars that he did "not accept what (the police) showed him", arguing that "the film has been cut".

Interior Minister Ahmad Vahidi said Saturday he had received reports that the emergency service had arrived "immediately" at the scene.

"Mahsa apparently had previous physical problems and we have reports that she had undergone brain surgery at the age of five," Mr Vahidi said.

Her father, however, "insists that his daughter had no history of illness and was in perfect health", Fars reported.