When 200 British Muslim imams declared earlier this month that they would refuse to say funeral prayers for the perpetrators of the recent terrorist attack near London Bridge, their statement marked a striking and unprecedented rejection of terrorism.

“We don’t take this matter very lightly,” says Qari Asim, an imam in the northern city of Leeds who signed the declaration. “But … we believe that the terrorists should not be accepted in our community, [either] in life or in death. We are trying everything possible to deter people” from violence.

The move signaled a significant change of tack. For many years, the almost universal reaction among European Muslims to the rising tide of jihadi violence has been to disclaim any responsibility on the part of Islam and the Islamic community.

But the attacks in Manchester and London “have shaken the Muslim community [in Britain] very deeply,” says Ziauddin Sardar, a London-based scholar of Islamic history. Now, Muslim leaders are beginning to tentatively acknowledge that their communities cannot shrug off all liability for the recent spate of terrorist attacks across Western Europe.

“Our first task is to own up and acknowledge that these people emerge from the Islamic community,” says Mr. Sardar.

'It is not enough to condemn'

The imams’ decision not to bury the terrorists came after British Prime Minister Theresa May, speaking in the wake of a knife attack near London Bridge that killed eight people, said that there was “far too much tolerance of extremism in our country.”

But European Muslims, divided between many schools of thought and traditions, are unlikely to unite around a single approach to terrorism.

Demanding that Muslims address the scourge in the name of Islam “would imply that Muslims are potentially terrorists and we don’t accept that premise,” says Imran Shah, a board member of the Islamic Society of Denmark. “We will not accept orders from someone pointing his finger at us saying ‘this is your fault.’”

But British Communities Minister Sajid Javid, himself a Muslim, argued in a recent op-ed article in The Times that British Muslims bear a “unique burden” to tackle extremism. “It is not enough to condemn. Muslims must challenge, too,” he wrote. “We can no longer shy away from those difficult conversations.”

Naz Shah, a Labour party member of Parliament from Bradford in northern England, says that her Muslim constituents have overcome their reservations and that “they are having conversations about empowering communities” to face up to extremists. “We are talking about this amongst ourselves,” she adds.

But she rejects the idea that Muslims tolerate terrorism. The Manchester suicide bomber, Salman Abedi, had been reported to police by friends and neighbors on three separate occasions, she points out.

In France, too, the tone of the debate is changing, says Rachid Benzine, a member of a government commission studying imams’ education. “In the past people were saying that terrorism either had nothing to do with Islam, or everything to do with Islam,” and nothing to do with adolescent rootlessness, or Western policy in the Middle East, or social discrimination at home, or other contributing factors, he recalls.

Now, he says, “there is a recognition that jihadism is a product of both international problems and of the way Islam has been ideologized.”

The danger of radicalism

Though Muslim public intellectuals may think like that, many preachers in French neighborhood mosques “are hesitant to criticize” extremists “because they are afraid of stigmatizing the whole religion,” explains Mr. Benzine. “But there is no way around this; they have to do it.”

In Britain, Mr. Asim says, Muslim leaders are taking up those cudgels. “We used to ignore those verses” in the Koran that urge Muslims to kill non-Muslims – the ones that Islamic State preachers seize on – he says. “But we need to talk about them and explain to students that they are part of the Koran but that they do not apply today because they refer to a particular historical context that is not the same today.”

But a knowledge of Islam is not enough to combat extremism, suggests Dounia Bouzar, founder of the first counter-radicalization program approved by the French government. Imams are not trained to spot the warning signs, she worries, and they may in fact be the least well placed to help someone in the process of being radicalized.

“The first thing jihadi recruiters do is cut a young man off from his mosque and his community, so that they are his only interlocutors,” Ms. Bouzar says. “It makes him easier to brainwash.”

Bouzar is trying to spur debate among French Muslims about the growing role that quietist, non-violent Salafists play in Muslim society. Though they reject jihad, they way they spurn music, art, sport, and human government, forcing adherents to choose between God and the world, cuts them off from the rest of society, she says.

Across the English Channel, Sara Khan, founder of the anti-extremist Inspire movement, has reached similar conclusions.

All extremism that promotes hatred of the other, she says, "even non-violent forms, has the potential to create mood music for terrorism."

Muslim and non-Muslim society “has chosen to be tolerant of intolerance,” Ms. Khan argues. “We need a sea change in attitudes,” she says. “We need to be upfront and honest in the Muslim community about violent and nonviolent extremists.”

Recent years have seen a growth in local Muslim groups ready to challenge extremists, she says, “but they are still in the minority, and people involved in counter-radicalization are called government stooges.”

'False news won't be combated with silence'

There is certainly a good deal of resentment among ordinary British Muslims about the government’s anti-terror strategy, known as Prevent, which legally obliges citizens to report any suspicions they may have about their neighbors.

“It turns people against each other,” complains Ms. Shah, “instead of encouraging them to have braver conversations” about how to curb terrorism.

At the Makkah mosque in Leeds, Imam Asim believes “it is absolutely fair to ask people to report” their suspicions. “Nobody wants [extremists] amongst us,” he says. They “are abusing our faith. No one wants to lose a young member of the community, and each attack gives rise to anti-Islamic hatred.”

Islamophobia is a growing problem in Britain. Anti-Muslim attacks have risen nearly fourfold since 2013, and 10 days ago a man drove his van into worshipers emerging from Ramadan prayers at Finsbury Park mosque, killing one of them.

“We must accept that Islamophobia is used by extremists to fan their narrative,” says Ms. Shah. “We can condemn [jihadi attacks] till we are blue in the face, but in the end it’s about … seeing that they are a threat to us.”

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In Copenhagen, Mr. Shah agrees. “The false news that is being spread around needs to be combated,” he says. “And it won’t be combated with silence.”

• Courtney Traub in London and Sara Miller Llana in Copenhagen contributed reporting to this article.