Online jihadist propaganda attracts more clicks in the UK than any other country in Europe, a report has found.
Britain is the fifth-biggest audience in the world for extremist content after Turkey, the US, Saudi Arabia and Iraq, Policy Exchange's study said.
The think tank suggested the UK public would support new laws criminalising reading content that glorifies terror.
The government has told internet companies like Facebook and Google to do more to remove jihadist material.
Former US military chief General David Petraeus, who wrote a foreword to the report, said efforts to combat online extremism were "inadequate".
He said the bombing of a London Tube train last week "merely underscored once again the ever-present nature of this threat."
"There is no doubting the urgency of this matter," he said.
"The status quo clearly is unacceptable."
The report suggested new laws to criminalise the "aggravated possession and/or persistent consumption" of extremist ideology - but not to criminalise someone who "stumbles across" jihadist content.
It said images of child abuse were approached in a similar way, with tougher penalties for the most serious cases.
Under section 58 of the Terrorism Act 2000, it is currently an offence to possess information that could assist a would-be terrorist, but not material which glorifies terrorism.
Policy Exchange surveyed 2,001 adults in the UK, finding 74% of people supported new laws to criminalise the "persistent consumption" of extremist material online.
'Vast' online presence
Its 130-page report found IS produces more than 100 new articles, videos and newspapers in a week - saying any decline of the terror group in the online space had been "significantly overstated".
"For at least a year, the production of content has continued despite the death of key figures, loss of territory and ongoing fighting," it said.
The jihadist group has retreated from territories it has seized in the Middle East, following pressure from a number of Iraqi and Syrian forces.
IS, also known as Daesh, disseminated its online propaganda across a "vast ecosystem" of platforms, the report found - including file-sharing services, encrypted messaging platforms and social media websites, as well as Facebook, Google and Twitter.
Internet giants say they have made efforts to clamp down on extremist content, with Google describing online extremism as a "critical challenge for us all".
Facebook said it was working "aggressively to remove terrorist content" from its website, and had developed a shared industry database of "hashes" - unique digital footprints - which catalogues violent extremist videos or images.
Twitter said that terrorist content had no place on its platform.
Home Secretary Amber Rudd said: "We know that Daesh pose a threat online and this report helps to highlight the scale of the issue.
"I have made it crystal clear to internet bosses that they need to go further and faster to remove terrorist content from their websites and prevent it being uploaded in the first place."
The report suggested the UK government introduce a "sliding scale" of measures to bear down on internet companies - including giving the proposed new commission for countering extremism powers to oversee the removal of online content.
Policy Exchange said 74% of Britons surveyed thought big internet companies should be more pro-active in locating and deleting extremist content.
Martyn Frampton, Policy Exchange's co-head of security and extremism, said governments and security services were playing a "fruitless game of whack-a-mole" by focusing on removing individual pieces of content.
"If the internet companies won't do what their customers want and take more responsibility for removing this content, then government must take action through additional regulation and legislation," he said.
Mrs Rudd added: "The internet cannot be used as a safe space for terrorists and criminals, and industry need to ensure that the services they provide are not being exploited by those who wish to do us harm."