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”.

Saurav Das, a CJP spokesperson, said: “It’s a Band-id. That’s not giving us accountability. There have been zero convictions so far all these years so this measure is of no use.”. A core demand of the movement is the resignation of the education minister, Dharmendra Pradhan, along 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.

Renuka Chowdhury, an opposition MP with the Congress party, told the India Today news channel that no one could fathom why Modi would not sack Pradhan. “Honestly is this one minister more valuable, more important than restoring harmony and order to the country?” he said.

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 resignation.

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 people in other cities across India have also voiced their support for the CJP’s demands by holding their own demonstrations. There were three consecutive days of demonstrations in Mumbai. Chennai, Indore, Hyderabad, Patiala – also had solidarity protests.

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.

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.

The health minister, JP Nadda, accused Gandhi of trying to make political gains from the protest. “Your concern is not to improve the education system, enhance the quality of education, or ensure justice for students. Instead, you are trying to use students’ issues and their demands for political mileage. You are trying to derive political benefit from them, and that is unfortunate,” he said.

CJP leaders were deeply disappointed when their meeting on Monday with Nadda – the first in over 60 days – resulted in nothing. Its founder, Abhijeet Dipke, has since said that if ministers want to speak to them, they will have to come to the protest site and talk to them in public.

Many in India are asking why Pradhan will not resign. Anish Gawande, an opposition politician, said Modi regarded backing down as weakness, while the prime minister no longer had the absolute majority he had in parliament in his first two terms.

“He is facing global headwinds that prevent him from executing any major reforms in agriculture, energy, the economy. The unemployment rate is climbing to levels not seen in three decades,” he said. “A show of weakness now will give an entry point for the opposition to move in and build a similar momentum around other issues […] Before you know it, things have spiralled out of control.”