The availability of psychoactive drugs was also a concern following a spate of hospital admissions.
Progress had been made since the last report but concerns remained, the chief inspector of prisons said.
The National Offender Management Service said it would "work hard to improve".
HM Inspectorate of Prisons (HMIP) carried out the inspection last autumn during 29 September and 3 October.
Inspectors said levels of violence had risen since the last inspection and "were now considerably higher than in similar prisons".
They said not enough was being done to "investigate individual violent incidents" and the prison was not dealing with perpetrators and victims.
Concern was also raised over the "availability of new psychoactive substances such as 'Spice'" - a substance which is a synthetic cannabinoid and acts like cannabis.
The inspection report cited seven "Spice-related emergency admissions to hospital" in the past six months.
But "the governor and staff had made huge efforts to respond to recommendations," HMIP said in its report.
Inspectors said they were "pleased" to find "a marked improvement in the staff culture and attitudes of staff since 2013".
They also said mental health provision was "very good", with prisoners at risk of suicide and self-harm feeling "well-supported".
But although "considerable progress" had been made, the prison still required support to make further improvements.
Chief Inspector of Prisons Nick Hardwick said: "HMP Bristol has come a long way in a relatively short time since the last inspection.
"The distance the prison has already travelled has involved real determination, hard work and resilience from managers and staff which needs to be recognised.
"It is important they now receive support to make the further improvements required."
Michael Spurr, from the National Offender Management Service, said: "There remain a number of key areas for improvement, and the prison will continue to work hard to address these, supported by 10 new prison officers who will start later this month."
The prison was previously criticised following an unannounced inspection in May 2013 when concerned inspectors said they would "return earlier than usual".
HMP Bristol is a category B local prison holding young adult and adult men.
Levels of violence have "risen sharply" at Bristol Prison, with not "enough being done to protect some vulnerable prisoners", a report has found.