Labour’s shadow foreign secretary made the claim after at least seven people were killed and 48 more injured after three terrorists launched their deadly attack in London, on Saturday night.
Speaking on BBC’s 5Live, Ms Thornberry said Labour would ensure the anti-radicalisation programme Prevent worked “as well as it possibly can” to stop further attacks in Britain.
When challenged over 3,000 so-called jihadis living in the country and potentially alienating the Muslim community policing them, Ms Thornberry added: “No, I think that we’ve heard, haven’t we, in relation to the Manchester murderers and the London murderers that certainly it has been suggested that there were attempts made to contact the authorities and say there were concerns about particular individuals.
“We can’t be expecting the Muslim community in this country to police itself, it has to work with the authorities so we need to make sure there are better connections, and trusted connections – that does take sensitive work and focus.”
Emily Thornberry has been slammed after claiming "Muslim communities" cannot police themselves
One Twitter user described Jeremy Corbyn’s shadow cabinet as “dangerous”, as the shadow foreign secretary was slammed over her defence of the Labour leaders anti-terror policies.
He wrote: “Emily Thornberry on 5Live yet again defended Corbyn’s past policies, this time on anti-terror.
“The man is a danger to us all.”
Commenting on Labour’s plans to put more police on the street, another said: “More police officers more important than targeting UK jihadis it seems. Flawless logic.”
A third added: “I was going to vote Labour but after hearing you more worried about upsetting communities than people being murdered? Unsure?”
Others were left furious that Ms Thornberry was sent instead of foreign home secretary Diane Abbott to comment on the domestic issue.
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A Twitter user said: “Diane Abbott is that useless that Emily Thornberry is sent to interviews over domestic issues!”
Another tweeted: “Emily Thornberry is struggling to explain why Diane Abbott is not commenting on the London terror attack.”
Ukip home affairs spokeswoman Jane Collins said that Labour "needed to stop making excuses for Muslims in Britain".
She added: "Why does Emily Thornberry not think that the Muslim community should consider itself part of the rest of the country and report people who wish to kill and maim?
"The Casey Report on integration in the UK highlighted the failures in integration and made many recommendations and said that governments have failed for more than a decade to ensure that social integration in the UK has kept up with the 'unprecedented pace and scale of immigration' and have allowed some local communities to become increasingly divided.
"A report from the Gatestone Institute found that more than 100,000 British Muslims sympathise with suicide bombers and people who commit other terrorist acts and only one in three British Muslims (34 per cent) would contact the police if they believed that somebody close to them had become involved with jihadists.
"Clearly we have a massive problem in Britain today – a divided society which the policy of multiculturalism has perpetuated. Much of the blame for this lies with the Labour Party: here in Yorkshire we know only too well the consequences of that party 'not wanting to rock the multicultural boat'.
"As a country, we must live by one rule of law: no exceptions, no excuses."
London terror raids Sun, June 4, 2017
Police have raided a block of flats in Barking in connection with last night's London Bridge terror attack.
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As police continue their investigations following the June 3 terror attacks, a block of flats in Barking is raided for possible suspects
Labour have come under significant pressure since the attack on London Bridge and Borough market, with Mr Corbyn’s opposition to ‘shoot to kill’ being challenged.
Ms Thornbery in a separate interview with Sky News said her party’s leader did back the policy despite some confusions about his views on the issue.
She said: “When it comes to defending people if somebody is coming up to you with a gun, it is absolutely within the law and indeed the duty of a police officer to kill that persons it if means it will stop a threat to life.
“What I support is police officers having the powers and it being entirely clear that they, of course, must always make a safe… and if that means killing someone in order to stop more people being killed, of course, that is the case, and, of course, that is what Jeremy believes in.”