MPs have asked how new rules aimed at stopping children seeing pornographic content will affect websites such as Twitter.
The Digital Economy Bill will lead to sites that do not ask for age verification being blocked.
It received an unopposed third reading from MPs on Monday evening, meaning it is set to become law.
But John Whittingdale, the Tory MP who first tabled the bill in July, is not convinced it will solve the problem.
"One of the main ways in which young people are now exposed to pornography is through social media such as Twitter, and I do not really see that the bill will do anything to stop that happening," Mr Whittingdale said.
Twitter's rules ban the use of "pornographic or excessively violent media" in profile or header images but allow posts to contain explicit adult content.
The amount of pornographic content on the platform has grown since June, when Twitter allowed users to post videos lasting up to two minutes and 20 seconds.
The Digital Economy Bill contains measures to bring in age verification for pornographic sites and withdraw payment services from sites that do not comply.
In the latest version of the proposed legislation, the government has given the British Board of Film Classification (BBFC) new powers to enforce the rules.
The bill's focus is on tackling sites involved in the commercial provision of pornographic material.
Digital Minister Matt Hancock acknowledged that dealing with sites where users could add material unchecked was more difficult.
Twitter's use among youngsters is declining, that's according to a recent report from the regulator Ofcom. However, that might change if access to pornography either explicit material is closed off by the Digital Economy Bill's provisions.
"I appreciate that there is a big challenge in stopping those who really want to access porn online, but all the evidence suggests that children's first interaction is often by accident," Mr Hancock said.
"We are legislating to prevent as much as possible of that inadvertent viewing by those who are not desperately actively seeking to do so.
"I appreciate that the bill is not a utopia, but it is a very important step forward."
Maria Miller, MP for Baskingstoke, reminded MPs that a few years ago Twitter had said it could not take down visual images of children being abused but now there was a code of practice in place to handle this.
"Surely where there is a will there is a way," said Mrs Miller.
"[Mr Hancock] has already proved that he can make significant progress, so should he not put more pressure on organisations like Twitter?"
"'Yes,' is the short answer," replied Mr Hancock.
"The bill does so, and we will best achieve that pressure by delivering on its proposals and then working with the platforms on the issue of platform-based pornography, because that is a much more difficult technical nut to crack."
Children's charity NSPCC would like to see industry-wide minimum standards for child safety on the internet.
"This would mean children will be able to go online safely without being exposed to porn and other inappropriate content, such as violent videos." a spokesman said.
"It's good news that the age verification measures we called for in the Digital Economy Bill are now on their way to becoming law.
"However, there are lots of other online spaces, including social media, where children will still be able to see pornography."
Twitter declined to comment.