Under new powers introduced by the Online Safety Act and passed under the previous Tory government, Ofcom will require internet companies to conduct stricter age verification methods to check whether a user is under 18.
A new code of practice, to apply from 25 July, will also require platforms to change algorithms affecting what is shown in children's feeds to filter out harmful content.
At the last election, Labour committed to "build on" the previous government's law and consider further measures to keep children safe.
But it is yet to publish fresh legislation of its own, with ministers arguing the existing set of new regulations need to be rolled out first.
Speaking to Sunday with Laura Kuenssberg, Ian Russell, whose daughter Molly took her own life at 14 after seeing harmful content online, said the new rules should mark the "biggest moment in online safety" since the arrival of social media.
But he added the "proof of the pudding is in what happens," adding he thought Ofcom could go further than it has done within the legal powers it has acquired.
He also argued that the regulator should be prepared to "push back" against ministers over "weaknesses" in the legislation.