The government is working on a new code of practice for social media companies to tackle bullying, intimidating or humiliating online content.
Cyberbullying, trolling and under-age access to porn would be targeted in an attempt to make the online world safer.
Facebook and Twitter could be made to pay for action to tackle the "undeniable suffering" the internet can cause, culture secretary Karen Bradley said.
This comes after a poll commissioned by the Headmasters’ and Headmistresses’ Conference of leading independent schools found that 28 per cent of pupils had suffered online abuse from people they knew. The same proportion said that they had received abusive comments online from strangers.
The government has also today confirmed plans to make relationship lessons – including online safety – compulsory in schools.
Social media companies will face an industry-wide levy to fund measures to tackle online harm.
The culture secretary defended making the levy voluntary, despite the Tory general-election manifesto promising legal powers to enforce such a move.
She told BBC Radio Four's Today programme: "I don't rule out legislating if that's what we need to do but I hope we can do it on a voluntary basis working with the companies.
"It's not backing away at all. It's saying: 'What is the best way to do this?'"
Ms Bradley said that the opportunities of the internet need to be embraced, adding: "But we have to do it in a safe way, and we have to it in a way that respects content as much as anything else and that's one of the issues about the status of internet companies and social media companies."
Monitoring online bullying
The proposals outlined in the internet-safety Green Paper also include an annual internet-safety transparency report, intended to monitor on online abuse.
Support would be given to digital start-ups to make sure they build safety features into new apps.
It highlighted research from the UK Safer Internet Centre that found 64 per cent of 13- to 17-year-olds have seen people posting offensive images or videos.
Opposition parties called for more detail and stronger policy. Shadow culture secretary and Labour deputy leader Tom Watson said: "We're pleased the government has accepted Labour's call for compulsory sex and relationship education in schools, including online safety education, as well as for codes of practice for social-media companies to prevent abuse.
"But this announcement is short on detail. The government needs to say more about who exactly will pay the proposed levy, how much they will pay and how it will be spent.
"They need to explain what transparency information they will be asking social media companies to provide."
'Power, information and resilience'
Anne Longfield, the Children’s Commissioner for England, said: "Government, parents, teachers, and the internet and social-media giants all have a responsibility for making sure children have the power, information and resilience they need to make the most out of the fantastic opportunities in today’s digital world.
“It is right that the multi-billion dollar companies who play such a huge role in the lives of our children should be doing much more to ensure all children, particularly the most vulnerable, are safe and have the information they need.”
But, she added, more could be done: “I will continue to call on the social-media giants to give children the tools they need to thrive online and to explore some areas further, such as the introduction of a digital ombudsman to act as a mediator between children and social-media companies."