Major reforms of the libel laws will also see internet service providers (ISPs) given greater protection from being sued if they help to identify so-called trolls.
Would-be claimants will have to show they have suffered serious harm to their reputations, or are likely to do so, before they can take a defamation case forward.
It comes after a mother who was targeted by online trolls won backing from the High Court to have her tormentors’ identities disclosed.
Nicola Brookes faced “vicious and depraved” abuse on Facebook after she posted a comment supporting former X Factor contestant Frankie Cocozza when he left the show last year.
The Defamation Bill was debated in the House of Commons on Tuesday.
But Scott Freeman, who set up the charity Cybersmile when his daughter became a victim of trolls, said these reforms do not go far enough.
“It needs to be a criminal offence, there needs to be legislation covering cyber-bullying. It’s no good with civil lawsuits,” he told Sky News.
“It needs to be addressed. It’s one in three children now throughout the UK [who experience cyber-bullying], and it’s affecting adults. It’s terrible and civil lawsuits are not enough. They’re unrealistic.”
Justice Secretary Ken Clarke said the Government wants a libel regime for the internet that makes it possible for people to protect their reputations but also ensures that information online cannot be easily censored.
“As the law stands, individuals can be the subject of scurrilous rumour and allegation on the web with little meaningful remedy against the person responsible,” he said.
“Website operators are in principle liable as publishers for everything that appears on their sites, even though the content is often determined by users.
“But most operators are not in a position to know whether the material posted is defamatory or not and very often – faced with a complaint – they will immediately remove material.
Sky News political correspondent Glen Oglaza said the reforms have widespread support across the parties, so the Bill should make its progress through Parliament pretty quickly.
“What they’re saying is, enough is enough of this cyber-bullying and abuse that’s going on, on websites like Twitter and Facebook,” he said.
“It’s a difficult position because although in theory Facebook and Twitter are the publishers in a way that books and newspapers, and for that matter television companies are, they don’t have the same amount of control over what appears and therefore, it’s more difficult to prosecute them and sue them for libel because in some ways, they’re not responsible.”
The power to ask for anonymous identities to be revealed are intended to give individuals an avenue of redress should they be subject to defamatory comments online.
The idea is to enable people to approach ‘offenders’ and seek out-of-court settlements or other mediated resolutions.