LONDON — Thousands of people turned out to demonstrate against police brutality and racism in European cities like London and Paris on Saturday, after a week in which statues linked to slavery and colonialism were targeted across the continent and calls intensified for scrutiny of policing and of a history of racial discrimination.

As protesters on the continent have showed solidarity with those marching in the United States in the wake of the police killing of George Floyd in Minneapolis last month, they have also denounced their countries’ own problems and urged the authorities to address them.

The situation was especially tense in London, where far-right groups came into the center of the city to stage an angry and at times violent counterprotest. They clashed several times with the police, who had imposed restrictions on the marches because of concerns about the potential for violent exchanges with protesters backing Black Lives Matter and left-wing causes.

Videos shared on social media showed mounted police officers standing guard in Parliament Square in front of statues of Gandhi and Nelson Mandela, which had been covered to protect them from vandalism. Protesters in the square were seen threatening and punching police officers who tried to repel them.

The protests and counter-demonstrations came after a week in which protesters in Britain tore down a statue of Edward Colston, a 17th-century slave trader, in Bristol, and others scrawled the word “racist” on a Winston Churchill statue in Parliament Square.

The statue of Churchill, a usually revered figure in Britain who steered the country through World War II, was later covered to protect it from being vandalized again, a move that Prime Minister Boris Johnson called “absurd and shameful.”

In Paris, less than two weeks after 20,000 demonstrators assembled in front of a court to demand justice for Adama Traoré, a 24-year-old who died in 2016 after being arrested by the police, thousands gathered on Saturday to reinforce the message. Earlier in the day, thousands had also attended Black Lives Matter protests in Australia, with gatherings held in cities like Melbourne, Perth and Sydney.

A far-right group in Paris that hung a banner denouncing “anti-white racism” on Saturday was booed by protesters.

Reckonings are also being called for in Germany, although gatherings planned across the country on Saturday were on a much smaller scale that on previous weekends.

The leader of a German opposition party, the Greens, has called for a change of wording in the country’s Constitution to include a provision that no one should face discrimination on the grounds of “racism.”

Chancellor Angela Merkel and her justice minister signaled this week that they would be open to a public debate on the change, which would require support from two-thirds of Parliament.

Across Europe, the authorities have walked a thin line in trying to quell public anger while also urging people to stay home in order to prevent the spread of the coronavirus, which has killed more than 167,000 people in Western Europe and dragged the continent into its worst economic slump since World War II.

The police in Paris reminded the public that gatherings of more than 10 people remain forbidden, and they asked businesses and restaurants to close near the Place de la République, where thousands gathered on Saturday.

The police also barricaded streets leading to the plaza, a move reminiscent of tense episodes that occurred in 2019 and early this year during the “Yellow Vests” demonstrations over a proposed pension overhaul in the country.

In London, the police urged people to avoid demonstrating in the city and set a 5 p.m. deadline for all protesters to leave the defined demonstration routes.

“I absolutely understand why people want to make their voices heard,” Commander Bas Javid said in a statement. “But the government direction is that we remain in a health pandemic, and people are asked not to gather in large groups.”

The protesters’ demands have met in some areas with resistance from police forces, who argue that instances of racism by their members have been isolated episodes.

After France’s interior minister on Monday promised “zero tolerance” of racism in the police and proposed a ban on chokeholds — a tactic that is increasingly being prohibited in the United States as protests there continue — the French police responded with a demonstration of their own on Friday and threw down handcuffs at stations across the country.

Dialogue between protesters and the authorities in Europe has been scarce. When demonstrators in Britain called on their country to acknowledge its racist and imperialist history, Mr. Johnson accused them of trying to “edit or censor our past.”

But there are also signs that politicians in Europe are heeding protesters’ calls.

Sibeth Ndiaye, a spokeswoman for France’s government, said on Saturday that the country had not brought an end to racism and that discrimination should be discussed more openly. In an opinion piece in the newspaper Le Monde, Ms. Ndiaye — who was born in Senegal and moved to France as a teenager — said the country needed to tackle “the reality of the suffering” that minorities experience.

Several celebrities have lent more direct support to protesters.

In Britain, the rapper Stormzy promised to give 10 million pounds, about $12.5 million, to anti-racism causes over the next decade. And in France, celebrities like the actor Omar Sy and the humorist Fary have called for a nationwide reckoning over police brutality.

Iliana Magra reported from London, and Elian Peltier and Constant Méheut from Paris. Melissa Eddy contributed reporting from Berlin.

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