Daryl Hargrave, who had a history of mental health problems, was found hanged in his cell in July 2015.
He had been on remand for six days, an inquest in the city heard.
The jury found there was a failure to provide treatment for psychosis, and a decision not to put him under constant supervision also contributed to his death.
Central and North West London NHS Foundation Trust, which manages healthcare at HMP Winchester, apologised for its failings and "the distress this process has caused the family".
It added improvements had already been made, including all mental health referrals now going to one point of contact and defibrillator training for all clinical staff.
Assistant coroner Karen Harrold said she had concerns about the risk of future deaths at the prison and felt further steps were needed.
Mr Hargrave's death was one of three self-inflicted deaths at HMP Winchester in two months during 2015.
Following an unannounced visit in 2014, inspectors described the prison as "insufficiently safe" with ineffective anti-bullying measures.
After the inquest, Mr Hargrave's mother Nicky said: "All we can ask is that lessons are learned and for the prison to get more resources and more training."
Solicitors representing the family said Mr Hargrave, from Gosport, had shown "clear symptoms" of psychosis days before his death and had told staff he had "demons in his blood, telling him to harm himself".
The jury heard he was arrested for allegedly being involved in a fight, and had been in prison for six days when he died.
His mother said he had struggled with addictions and had a history of mental health problems and self-harming.
The Ministry of Justice (MoJ) has not yet commented.
Neglect contributed to the death of a 22-year-old man at Winchester prison, an inquest jury has concluded.