A new Commission for Countering Extremism proposed by Conservatives will have a remit to clamp down on "unacceptable cultural norms" such as female genital mutilation.
It will also act to ensure that women's rights are upheld in all of Britain's ethnic and religious communities, Theresa May has said.
Mrs May was spelling out further details of the commission proposed in the Conservative manifesto for the June 8 General Election, which will have the task of helping government identify policies to defeat extremism and promote pluralistic values.
Meanwhile, security minister Ben Wallace launched an attack on "duplicitious" social media companies which refuse to share data with government on potential extremists, while being ready to exploit their access to innocent people's details in order to make money.
The Prime Minister described her "harrowing" visit to Manchester Children's Hospital to meet victims of Monday's suicide bomb attack on a pop concert in the city, saying the atrocity was "particularly sickening" because it targeted young people.
Recalling one girl who she met, Mrs May told the Sunday Express: "The young girl was being incredibly brave about what must have been an absolutely traumatic experience for her.
"I also saw her parents who rushed to the scene as soon as they heard there had been this terrible attack with their daughter caught up in it.
"What was clear to me was the intense trauma of this. There are the physical injuries that people go through and that have to be treated in hospital.
"But there is also the impact of the memories of what you have seen and what you have been through. It must have been horrific for parents to receive that phone call."
Experts have suggested suicide bomber Salman Abedi's decision to target a pop concert attended by teenage girls may have been driven in part by radical Islamists' opposition to the freedoms enjoyed by women in Western societies.
And the Prime Minister made clear that the commission will be expected to stand up for women's rights as extremism often goes hand in hand with poor treatment of women and girls.
A Conservative government would give the commission teeth by making it a statutory body with a legal responsibility to identify extremism and support people and organisations in resisting it, she said.
It will advise the government on new criminal offences and policies to defeat extremists and will offer assistance to public sector bodies in identifying extremism and stamping it out.
It will also be required to promote Britain's "pluralistic" values and to tackle non-violent extremist activities which undermine social cohesion and deprive individuals of the rights and expectations they are entitled to as part of British society.
Modelled on the Commission for Racial Equality, it will aim to mount a determined effort to root racism out of British society in the same way that officialdom took on racism from the 1970s onwards.
Mrs May said: "Britain is one of the world's most successful multi-racial, multi-religious and multi-cultural societies.
"But our enjoyment of Britain's diversity must not prevent us from confronting the menace of extremism, even if that is sometimes embarrassing or difficult to do.
"Extremism, especially Islamist extremism, strips some people of the freedoms they should enjoy, undermines the cohesion of our society, and can fuel violence. And it can be especially bad for women.
"There is clearly a role for government in tackling extremism where it involves behaviour that is or ought to be criminal.
"But there is also a role for government to help people and build up organisations in society to promote and defend Britain's pluralistic values, and stand up to the extremists who want to undermine our values and impose their twisted beliefs onto the rest of us. That is what this plan is all about.
"Enough is enough. We need to be stronger and more resolute in standing up to these people."
Mr Wallace told the Sunday Telegraph: "Some of the (social media) names are actually helpful, but some of them are completely duplicitous.
"A lot of these companies survey how you and I live our lives online. They take that data and they sell it around the world to anyone from loans to soft porn.
"But when we, the state, say we have processes and warrants and we ask for that same type of data, we get this very twisted view of 'Ah, but that's surveillance'.
"My details are flogged by these companies around the world ... It is just capitalism. They are ruthless moneymakers."
The Conservative chairman of the parliamentary Intelligence and Security Committee, Dominic Grieve, said the fight against terrorism made it crucial to maintain close co-operation with EU law enforcement agencies like Europol after Brexit, even if this requires some continuing involvement with the European Court of Justice.
"Although our partnership with the US for intelligence sharing is extremely important, the fact is that the current terrorist threat is very much a European dimension issue.
"The Schengen database and knowing about who has moved where are all intimately dependent on European systems and we have got to try to remain in them," Mr Grieve told The Observer.
"I think it is going to be very difficult, and the government and Parliament will have to face up to the fact that it might turn out during the Brexit negotiation that some sort of mechanism for the European court is going to be necessary."