Avon and Somerset Police have urged people considering taking part in further protests in the city over the weekend to "reconsider for the sake of public health".
In a statement, chief superintendent Claire Armes said: "While Covid restrictions are in place, gatherings of any kind will only put our communities at risk.
"This pandemic has cost many lives and is still a significant challenge for our colleagues in the NHS. Anyone who chooses to flout the restrictions is playing a part in prolonging this difficult situation.
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"We fully understand the strength of feeling around the right to protest, but now is not the time. We're again asking people to please do the responsible thing and stay at home this weekend.
"Policing resources are in place to ensure we're able to keep our communities safe and maintain order."
The warning comes after violence erupted at the third Kill the Bill demonstration in Bristol on Friday, with 10 arrests made following what police called unacceptable "violent conduct".
Prime Minster Boris Johnson branded the violence towards police as "disgraceful" and said that police and the city had his full support.
Mr Johnson tweeted: "Last night saw disgraceful attacks against police officers in Bristol.
"Our officers should not have to face having bricks, bottles and fireworks being thrown at them by a mob intent on violence and causing damage to property. The police and the city have my full support."
Home Secretary Priti Patel also tweeted that it was clear that "thugs" had only been intent on causing trouble
She added: "I am disgusted by the disorder in Bristol and the violence being directed towards the police.
"I'm in no doubt the silent, law-abiding majority will be appalled by the actions of this criminal minority.
"Despite repeated warnings to disperse, it's clear these thugs were only intent on causing trouble."
Meanwhile, demonstrators took to streets across the country on Saturday for further Kill The Bill protests - including in Manchester, Bath and Sheffield.
Some 300 people initially joined a protest march through Bristol city centre against the Government's new Police, Crime, Sentencing and Courts Bill on Friday night, before the crowd swelled to more than 1,000 as tempers flared.
Police officers wearing helmets and holding shields moved towards the protesters after 10pm, after calling for the crowd to disperse due to Covid restrictions.
Avon and Somerset Police said glass bottles, bricks and eggs were thrown at officers and fireworks were launched at its mounted division, with one horse being daubed with paint.
Protesters had also shone laser lights into officers' faces, the force said.
The force also said it was trying to contact a journalist who claimed they had been assaulted by officers in Bristol on Friday night.
It came after a video was shared on social media of an officer appearing to use a riot shield to push back a Daily Mirror reporter.
In a statement on Twitter, the force said it was aware of the video, adding: "A free press is a cornerstone of our democracy and we fully respect the media's vital role in reporting events fairly and accurately."
Three of those arrested were also detained in connection with last Sunday's first march, which was followed by another violent demonstration on Tuesday night.
A large police presence was on hand during the demonstration, with horses and dogs used to help move the crowd back.
Prior to the arrests, protesters had been spotted dancing to music despite heavy rain, handing out flowers and chanting slogans such as "Who do you protect?" and "Justice for Sarah", in reference to the death of Sarah Everard.
Serving police officer Wayne Couzens, 48, from Deal, Kent, has been charged with her murder.
Rows of officers and vans blocked the protesters from Bridewell police station, the scene of last Sunday's violence.
The proposed Police, Crime, Sentencing and Courts Bill would give police in England and Wales more power to impose conditions on non-violent protests, including those deemed too noisy or a nuisance, with those convicted liable to fines or jail terms.
Bristol Mayor Marvin Rees said it was concerning how many people were travelling to the city to protest or "cause conflict".
In a statement on Facebook on Saturday, he added: "The question those engaging in the action should be asking is: is what I am doing advancing the cause I claim to be campaigning for?
"Many people protested peacefully but there are a number who refuse to go home and others who are here merely to cause conflict: the Bill itself is not their cause, it is their opportunity.
"If the protests are meant to reduce the likelihood of the Bill, then the actions of some of these protestors are politically illiterate and strategically inept."
The first demonstration last Sunday began peacefully with around 3,000 attending, but descended into a riot when some 500 people marched on Bridewell police station.
Twelve people have been arrested in connection with that protest, in which officers were attacked, police vehicles were set on fire and the windows of a police station smashed.
The force later retracted claims two officers suffered fractures in Sunday night's riot.
Tuesday night's protest resulted in 15 arrests.