The protests were organised following the Southport tragedy last week where three young girls were killed and eight more children and two adults were injured in a knife attack.
There were some incidents of violence during the protests with objects (including glass bottles) and punches thrown.
The demonstrations moved to outside of a hotel housing asylum seekers, where the counter-group formed a human wall to prevent the protesters from entering.
Chief Inspector Vicks Hayward-Melen said: “This was a dynamic police operation and significant policing resources were put in place.
“We have a long history of facilitating peaceful protest in Bristol, but what we saw last night was not a protest and we will never accept such shameful scenes.
“Those responsible can expect a knock at the door from us soon – we will be relentless in identifying and bringing to justice those who brought criminal violence and disorder to the streets of Bristol [on Saturday]”
16 individuals have been arrested so far following the disorder. Offences include affray, racially-aggravated public order, animal cruelty, assaulting an emergency worker and violent disorder.
In a joint statement, Bristol’s political party group leaders said: “The violent scenes witnessed over the weekend were carried out by a small group of individuals who chose to express their views through fear and intimidation. This does not reflect our values as a city.
“We thank the emergency services, our community partners and council teams for their efforts to maintain public safety and respond to the need of those who required support. Their efforts ensured this weekend's events were contained and that communities continue to be supported.”