The Conservatives have pledged to give people the power to force social media companies to delete embarrassing content they posted as children.
The Prime Minster said sites like Facebook would be forced to wipe the records of users before they turned 18 on request.
Online companies will also be warned they face penalties if they fail to do more to stop children accessing harmful content.
Mrs May said the internet had brought "significant new risks" as well as a wealth of opportunities.
It comes as the NHS was left reeling from a massive cyber attack that hit some hospitals, GP surgeries and commissioning groups.
"The internet has brought a wealth of opportunity, but also significant new risks which have evolved faster than society's response to them.
"We want social media companies to do more to help redress the balance and will take action to make sure they do.
Home Secretary Amber Rudd told ITV News that while the social media and digital technology was "fantastic" for young people, they should have the right to have material posted before the age of 18 removed "just in case there might be some material there that might impact on their life in future".
Under the plans, social media firms will have to take action to stop search terms directing users to inappropriate sites.
The measure would also include hate speech and other sites that could be harmful to youngsters.
Age restrictions on apps to stop young children accessing damaging content will also be expanded.
Firms will be told to introduce a "comply or explain" system that means they take down content that has been complained about or explain the reason behind why they have not.
The plans also include widespread changes to make conducting business online simpler such as allowing a digital signature to be accepted on official forms such as contracts.
Liberal Democrat former Cabinet minister Alistair Carmichael said: "Government and technology companies must do more to find a real solution to problematic content online but having a government agency deciding what constitutes acceptable free speech isn't it.
"We need to be working with technology companies to address the problem of hate speech, not pretending it's an easy problem that can be solved with a press release."