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.

Carol Boardman sustained serious injuries after a collision with a pick-up truck as she was cycling in Connah's Quay, Flintshire, on Saturday.
She later died in hospital.
Ch Insp Darren Wareing said: "A full investigation is now under way, which will include a forensic examination of the scene, vehicle and bike.
"We have already spoken with a number of people, including the driver of the Mitsubishi pick-up, however I'd like to ask again if you were in the area at the time and witnessed the collision or any events leading up to it, to contact the Roads Policing Unit on 101 or via Crimestoppers on 0800 555 111 and quote RC1610 4385."
Chris Boardman said his mother was "the most positive, outgoing person you could ever hope to meet".
The 47-year-old, who won a gold medal at the 1992 Barcelona Games, tweeted a photograph alongside a tribute to his mother, who he said "never lost her love of the bike or competing".
He also wrote about her "generosity of spirit" and love of the outdoors, recalling fond childhood memories of walking and exploring as a family in north Wales.
"We love you Carol, may the wind be ever at your back," he said.

The Spanish-language TV network had cancelled its contract to broadcast the Miss USA pageant which is co-owned by Mr Trump.
Univision had said last week it would not air the pageant because of what it called insulting remarks about Mexican immigrants.
Mr Trump is running for the 2016 Republican presidential nomination.
Mr Trump has accused Univision of stifling his freedom of speech in order to boost the Democratic Party frontrunner, Hilary Clinton.
While announcing his candidacy earlier this month, he had accused Mexico of sending rapists and other criminals to the United States.
He also pledged to build a "great wall" on the US border with Mexico and insisted it would be paid for by Mexicans.
He later insisted he was criticising US lawmakers, not Mexican people.
US TV network NBC has also said it would not be airing the Miss USA and the Miss Universe pageants both owned by Mr Trump.
Responding to the announcement, Mr Trump said he would consider suing NBC.
Levels of violence have "risen sharply" at Bristol Prison, with not "enough being done to protect some vulnerable prisoners", a report has found.
[NEXT_CONCEPT]
North Wales Police has renewed an appeal for witnesses following a road accident in which Olympic cyclist Chris Boardman's mother died.
[NEXT_CONCEPT]
The US presidential candidate Donald Trump has filed a multi-million dollar lawsuit against Univision.