Narendra Modi has said he will set up “fast track courts” to expedite the punishment of corrupt officials who leaked examination papers,as India’s prime minister seeks to quell the youth-led “cockroach” protest movement that has rocked Delhi and challenged his government.

In his first public remarks on the movement, Modi said on X “Nothing is more important than the welfare and future of our youth … Those who try to harm the future of our youth will not be spared.”

The offer of faster justice than usual from India’s clogged judicial system, which can take decades to deliver a verdict, was swiftly rejected by the Cockroach Janta party (CJP), the satirical name the youth protest movement coined after India’s top judge called unemployed youths “cockroaches”.

“Dharmendra Pradhan must resign,” the CJP said, referring to the education minister. His resignation is a core demand of the movement, along with reform of the education system.

An estimated 50,000 people protested in Delhi on Monday, even though police denied them permission to march on parliament and later mounted baton charges and fired teargas. On Wednesday, more than 1,000 people remained camped out at the CJP’s protest site in central Delhi despite the gruelling heat and humidity. Many protesters had travelled for miles to be there.

Modi’s statement follows indications the government is softening its stand somewhat. Ministers have been at pains to say they are willing to talk to the CJP but the one thing they refuse to mention is Pradhan’s resigation.

The protest site at Jantar Mantar is overflowing with supporters eager to show their solidarity. Unable to fit into the enclosed area, the crowd has spilled outside on to the roads. Some sort out donations of food and water sent by well wishers, others hand out food, including to the police on duty, fan those wilting from the heat, or clear up litter.

Young men and women in other cities across India have also voiced their support for the CJP’s demands by holding their own demonstrations.

At a press conference on Wednesday, the opposition leader Rahul Gandhi escalated the rhetoric, demanding that apart from Pradhan resigning, Modi should apologise for failing to provide a fair education system.

Demonstrators hold anti-Modi placards during a protest. Photograph: Ashish Vaishnav/SOPA Images/Shutterstock

Gandhi said the country had seen 152 exam paper leaks over the past decade without resulting in a single conviction.

“Somebody, or a group of people, are destroying our education system, India’s most valuable asset, destroying the lives of our students. What does the government do? Zero convictions,” he said.

If anything, the movement is hardening its stance after Monday’s brutal crackdown by the police. CJP leaders were deeply disappointed when their meeting on Monday with the health minister, J. P. Nadda – the first in over 60 days – resulted in nothing.

Its founder, Abhijeet Dipke, has said they will not go to the government. If ministers want to speak to them, they will have to come to the protest site and talk to them in public, he said.