The PM said “enough is enough” and called for “difficult conversations” to weed out potential terrorists from UK communities, adding that Britain has been far too tolerant of extremism.
She also accused global technology firms of providing a “safe space” for Islamist ideology to breed.
The boss of British lobby group TechUK praised the companies for being “absolutely committed to working through an international forum to strengthen existing initiatives; improve identification and removal of terrorist propaganda; and promote counter-speech, empowering those with inclusive and positive messages”.
Earlier in the day May spoke outside Number 10 Downing Street in the wake of Saturday night’s attack on London Bridge and Borough Market, in which seven people were killed and 48 injured, many with life-threatening injuries.
The incident follows a suicide bomb which killed more than 20 people in Manchester last month, and an attack in Westminster in March which claimed the lives of five victims.
During her speech, May said “we cannot allow this ideology the safe space it needs to breed. Yet that is precisely what the internet, and the big companies that provide internet-based services, provide.”
The PM called for “international agreements that regulate cyberspace”.
May added: “We need to do everything we can at home to reduce the risks of extremism online.”
Campaigners at the Open Rights Group said the government’s focus on the internet and encryption was “disappointing” and risked “distracting [from] hard and vital questions” around the causes of terrorism.
“This could be a very risky approach. If successful, Theresa May could push these vile networks into even darker corners of the web, where they will be even harder to observe,” the group warned.
However, in a hard-hitting speech, the PM also promised to “review Britain’s counter-terrorism strategy to make sure the police and security services have all the powers they need.”
On Saturday night, armed officers fired an “unprecedented” 50 rounds at the three terrorists, who had driven a vehicle into pedestrians before attacking other members of the public with knives. All three were killed.
Corbyn also criticised May over the UK’s foreign policies.
“It is no good Theresa May suppressing a report into the foreign funding of extremist groups,” he said.
“We have to get serious about cutting off the funding to these terror networks, including Isis, here and in the Middle East.”
Police made 12 arrests in connection with the terrorist assault at London Bridge. The Met police’s counter terrorism unit made the arrests in Barking, east London yesterday morning, with searches taking place at addresses in the area.
London mayor Sadiq Khan will hold a vigil at 6pm this evening at Potters Fields Park beside City Hall, to honour the innocent people killed and hurt in Saturday night’s attack.
And Downing Street said there will be a nationwide minute’s silence held tomorrow morning to remember and honour the victims. In a statement, Number 10 said the minute’s silence would take place at 11am tomorrow.
• London Bridge underground and national rail stations will re-open this morning, Transport for London said last night. However, commuters will not be able to walk across the bridge, so are advised to travel to alternative stations such as Cannon Street.