The prison is taking action to address some of the issues raised
The high number of inmates at HMP Bristol is affecting its performance, according to prison inspectors.
A spot-check of the jail in March found there had been improvement since the last full review in 2005.
But some prisoners are having to share cells designed for one person and black and ethnic minority prisoners reported worse treatment than white inmates.
A prison spokesman said action was being taken to address some of the issues raised.
Anne Owers, Chief Inspector of Prisons, said: "Managers at Bristol had succeeded in reversing the decline we recorded at the last inspection in 2005.
"However, in spite of these efforts, the effects of continued population pressure meant that Bristol was not yet performing well enough in three crucial areas - safety, respect and activity."
While staff prisoner relationships were generally good, there was no effective personal officer scheme, and race relations work was significantly underdeveloped, the report said.
There was also said to be too little vocational training and, at any one time, a third of prisoners were unable to engage in activity.
However, health care had improved.
Phil Wheatley, Director General of the National Offender Management Service, said on behalf of the prison: "Bristol is a busy local prison and works hard to provide a safe, decent environment.
"I am pleased that the Chief Inspector has acknowledged the hard work by managers and staff in raising the standards since the last inspection.
"There have been significant improvements in resettlement and healthcare provision at the prison and the Inspector's recommendations on race equality and purposeful activity have been accepted and action is being taken to remedy these issues."