Police have opened an investigation after a broken skull, a shoulder blade and leg-bones were among suspected human remains discovered lying uncovered in a cemetery.

The bones include a partial skeleton and were found at the privately owned Tottenham Park cemetery in north London by a group of campaigners who fear that graves are being dug up and reused without consent amid a nationwide shortage of burial space.

More bones have been found by members of the Tottenham Park Cemetery Action Group who have relatives buried there. Police and a leading forensic pathologist have confirmed to the Guardian that some of the bones discovered in recent months are human remains. Tests are pending on others, according to Scotland Yard.

The cemetery is owned by an Essex-based company that runs two private cemeteries in London and charges up to £4,100 per plot. Tottenham Park is one of the largest Turkish-Cypriot burial grounds in the UK and part of it is leased as an Islamic cemetery.

While four out of five people are now cremated in Britain, demand for burials remain high among Turkish and Muslim communities that require it for religious and traditional reasons.

Concern about the management of the cemetery has been simmering for years. David Johnson, a medical engineer who has five members of his family buried in three neighbouring plots, claims he discovered in 2005 that two elaborate marble structures had been erected over two of the plots to memorialise two completely unconnected people who appeared to have been buried there.

He said he has not been able to get answers about what has happened to the remains of his loved ones. “What have they done with them? We think they have been dumped in a skip. I feel very angry to have this done to my family.”

Baroness Meral Hussein-Ece, a Liberal Democrat peer whose parents, brother, two uncles and cousin are buried in the cemetery, said: “We are worried that just as some graves have disappeared our relatives’ graves will disappear. We want to be able to protect this place so our families are safe.”

The Burial Act 1857 makes it an offence to remove buried human remains without a licence from the government, or permission from the Church of England if it is consecrated ground.

David and Valerie Johnson. David says members of his family have disappeared after marble structures were erected at Tottenham Park cemetery. Photograph: Christian Sinibaldi/The Guardian

Scotland Yard said a man has been interviewed under caution as part of its investigation, which began in May when the first bones were found. It said it was alerted to the discovery of more bones on 29 August and “a dismantled human-looking partial skeleton” on 14 September.

“Police will continue to liaise with local volunteer groups and the Tottenham Park cemetery to repatriate the [latest discovered] bones, once confirmed as human, at the location as soon as is practicable,” said a spokesman.

Previous government reviews have suggested that burial space in the UK could run out in 10 to 15 years and consideration has been given to digging out bodies, lowering the grave and putting the body back with room on top for another coffin. The practice has yet to be approved.

“I freaked out when I saw the jaw,” said one of the volunteers, who asked not to be named. “All the bones were jammed in between the fence and the wall. That was the fourth lot of bones we found in four months.”

Scotland Yard said that bones discovered at Tottenham Park cemetery in May and June were forensically examined and confirmed as human.

However, Tottenham Park Cemetery Limited said: “To the best of our knowledge some bones were found on land not owned or managed by Tottenham Park Cemetery Ltd.”

Asked to clarify if it meant that bones were found on the part of the cemetery leased to the Tottenham Park Islamic Cemetery Association, it declined to comment.

Tottenham Park cemetery, looking on to the area the where bones have been found. Photograph: Jill Mead (assisted by Sumi )/The Guardian

The company said: “The police have established that bones found on land owned and managed by Tottenham Park Cemetery Ltd are those of animals.” It also declined to comment on David Johnson’s concerns.

Many of the bones discovered were in a crevice beside a concrete fence bordering the Islamic part of the cemetery that still has plenty of space. They include bones identified from photographs by a forensic pathologist as human femurs, a tibia, several ribs, a shoulder blade, a lower jaw and parts of skull.

Several memorial stones in the cemetery are broken and others appear to have been moved from their plots. Some were propped against the fence close to where the bones were found and do not appear to be Islamic.

They included memorials for Mary Bullen who died on Christmas Day 1926 aged 104, two-year-old Paul Gloor who died in 1934, and Edward Martin who died in 1948.

The Tottenham Park Islamic Cemetery Association said on Tuesday there was ample space in its section of the cemetery and it had no interest in reusing existing marked graves.

“We can categorically state that our organisation, run by volunteers to offer a community service at minimum cost, has never removed any remains from a grave,” it said in a statement. “We condemn such actions in the strongest terms as disrespectful to the deceased and their families.”

The cemetery was opened in 1912 and includes the graves of 10 first world war servicemen, including 18-year-old Alfred Gigg who died of his wounds in 1918, as well as some Austrian and German internees.

The leader of Enfield council, Nesil Caliskan, said the condition and management of the cemetery was causing distress and alarm and that it was unacceptable problems have not been adequately addressed. “I am calling on the secretary of state for justice to exercise his power and to intervene immediately,” she said.

A Ministry of Justice spokesperson said it was working with Enfield to consider how best to address some of the issues relating to the maintenance of the cemetery.