HMP Woodhill in Milton Keynes had five suicides in 2015.
Relatives of Kevin Scarlett, who killed himself in 2013, claim there were missed opportunities to prevent his death and "nothing has changed" since.
Prison minister Andrew Selous MP said the jail launched a review into the way prisoners at risk were supported.
A Prisons and Probation Ombudsman report into the death also highlighted a number of concerns at HMP Woodhill.
Mr Selous said: "We make every effort to learn from each death."
Mr Scarlett, who was from Milton Keynes, was charged with armed robbery and remanded at Woodhill in January 2013.
He killed himself on 22 May that year, aged 29.
Read more on this and other stories from Buckinghamshire
Lee Jarman, his step-brother, said Mr Scarlett had a history of mental health problems.
"Staff were missing key indicators such as self-harming and he had made attempts on his life while in prison," he said.
"The family is angry because three years on they still haven't learnt any lessons about how to save lives."
Ministry of Justice figures showed 89 people took their own lives in prisons across England and Wales in 2015, with Woodhill having the highest number with five, followed by four each at Exeter, Ranby and Winchester.
The Howard League for Penal Reform said its research showed there needed to be more non-custodial sentencing for criminals who had not carried out violent offences.
Frances Crook, its director, said: "We are locking up a lot of fragile people who have mental health problems alone for hours and it is very dangerous.
"Overcrowding is so bad, staff simply don't have time to talk to people and build relationships with prisoners."
Mr Selous said: "Following the death of Kevin Scarlett, HMP Woodhill reviewed the way prisoners at risk are supported and have introduced a number of measures to help prevent other self-inflicted deaths, including improved staff training and a new task force."
The family of a man who killed himself at a jail with the highest number of suicides in England and Wales say the prison has not "learned any lessons".