One of the highest-achieving state schools in the country is among 23 secondaries and primaries where former pupils have come forward in the wake of the Jimmy Savile scandal to claim that they were sexually abused by teachers, the Observer can reveal.

A former pupil at Tiffin School in Kingston upon Thames, south-west London, has come forward to tell the police that he was repeatedly abused by a male teacher at the boys' school in 1975. It is understood that the alleged perpetrator of the abuse is still alive.

The scandal that erupted in 2012 over Savile has encouraged a wave of alleged victims to come forward to the police to report their own experiences as children. The Observer has learned that the Tiffin School, which recently featured in a guide to state schools in the high-society magazine Tatler, is one of four state schools where allegations of abuse have been reported to police and lawyers acting for victims in recent months. In one case, the state school – not Tiffin – has made a settlement on behalf of the victim which is covered by a confidentiality agreement. In another primary school, there is an ongoing police investigation.

In a recent case, at Hillside First School in Weston-super-Mare, Somerset, the headteacher was dismissed when it emerged that staff had witnessed a total of 30 inappropriate incidents involving one teacher, none of which were passed on to the local authority.

The teacher Nigel Leat, 51, had received just one verbal warning about his behaviour, leaving him free to abuse children as young as six between 1996 and 2010, until a mother finally reported him to police. In June 2011, Bristol crown court heard that Leat had abused five victims, some as young as six. The judge described Leat as a "paedophile of the most sickening order" and he was jailed indefinitely.

Liz Dux, a lawyer from Slater & Gordon solicitors, who are representing alleged victims of abuse at schools across the country, said it was a myth that sexual abuse was only being uncovered at private schools as people come forward following the Savile revelations. While some leading public schools – including those attended by Nick Clegg, Boris Johnson and Tony Blair – face accusations of covering up abuse for decades, there is growing evidence of similar criminality in the state sector, said Dux. "There has been an explosion in the number of reported abuse cases. These are not exclusively in the private sector."

Dux, who represents about 70 alleged victims of perpetrators other than Savile, added that some schools involved were not yet aware of allegations being made about their staff. "Many of the schools we are currently investigating may not know of the allegations at the moment because, once people have contacted us, we encourage them to go straight to the police," she said. "And we don't take further action until the police have concluded their investigation and a prosecution has taken place."

Where civil claims are made against state schools, the schools are expected to compensate victims through liability insurance. In some historical cases it is understood that the institutions struggle to uncover with whom they have a policy, leaving the taxpayer to foot the bill.

A place at Tiffin School is keenly sought after, with 1,863 boys sitting their last 11-plus test competing for 160 places.

In the private sector, 18 former pupils of Ashdown House in East Sussex – which was attended by London's mayor, Boris Johnson, and the Homeland star Damian Lewis – have claimed that staff committed "very serious abuse" on children aged between seven and 13 during the 1970s. The £23,000-a-year boarding school is a feeder school for Eton and Harrow. Victims at Clegg's former preparatory school, the £22,000-a-year Caldicott School in Farnham Royal, Buckinghamshire, are also suing for damages.

The Chorister School in Durham, the £18,000-a-year prep school attended by Blair, recently settled a series of sex abuse claims. Police investigated allegations by 23 former pupils that they had been abused by Canon John Grove, headmaster from 1957 to 1978.

The Tiffin School has declined to comment; the claims are unsubstantiated.