The bill puts a duty on tech giants like Facebook and Google to develop systems to identify and remove illegal material, as well as deal with content that is harmful to adults and children. Ofcom would enforce this as the regulator.
An open letter led by the NSPCC (National Society for the Prevention of Cruelty to Children) charity and signed by the parents of children who have been groomed expressed frustration that the long-delayed bill was held up yet again last month in its passage through parliament.
Addressed to Conservative Party leadership candidates Rishi Sunak and Liz Truss, the letter states: "It was upsetting to see opponents of the bill rejoicing at this delay and calling for the legislation to be scrapped altogether.
"The Online Safety Bill would begin to hold tech companies accountable for the harm we have suffered that should never have happened in the first place.
"It has been years in the making and would finally turn the tide towards keeping children safe in a place where they spend so much of their lives."
Critics of the bill have said its proposals to restrain the publication of "lawful but harmful" speech, effectively create a new form of censorship of otherwise legal speech.
Truss, who is likely to win the leadership competition, has also expressed similar concerns over the bill and its possible impact on freedom of speech.
During hustings organised by GB News earlier this month, she said: "What I want to make sure the Online Safety Bill does is protect the under-18s from content that is damaging.
"Where it's about adults being able to speak freely, they absolutely should be, and it should be the same online as offline. That's a really important principle, and I'll make sure the Online Safety Bill does reflect that."
The bill's latest delay was called for so that it could be implemented once the new Prime Minister is in place following the summer recess of parliament.
The letter continues: "This level of harm against children would not be accepted in any other industry, which is why the next Prime Minister must make the Online Safety Bill a national priority and pave the way for urgent regulation to begin.
"Should you become the next Prime Minister, we urge you to keep the promise made to children and families and deliver a robust Online Safety Bill in full and without delay.
"Any watering down of the bill would be unacceptable and break the commitment made to children and families in the Conservative Party manifesto to deliver the strongest possible protections for children online."
