New parliamentary reports call for the addition of scams and offences such as sending unwelcome sexual images or encouraging violence against women and girls.
It stated that a named senior manager of the tech giants should be held personally responsible for any failures.
The report's authors stated that they needed to "call time on the Wild West online".
Chairman of the joint committee that issued the report, Damian Collins said: "What's illegal offline should also be regulated online."
"Big tech has been allowed to be the land of the lawless .... for too long. The era of self-regulation has ended for big tech."
More fines for new offences
Online Safety Bill is considered one of the most ambitious attempts to regulate online content to date. It could have global consequences.
In May's first draft, a duty of care was placed on large social media sites to remove illegal or harmful content and protect children. It was mostly left up to tech giants to police the first draft, published in May. Ofcom had oversight. But the parliamentary report demands that Ofcom set more specific standards and has greater power to fine and investigate large tech companies.
There are many recommendations that were made in the 191 pages.
All pornography websites have an explicit obligation to ensure that children can't access them
Fraud and scams, such as false advertisements designed to trick users, should be addressed
The bill should not only cover content but also "the possible harmful effect of algorithms".
It should be extended to include paid advertising such as scams.
Based on the Law Commission's proposals, the report recommends the creation of a broad range of new criminal offenses. These should be included in the bill.
Propagating or "stirring up violence against women" or based on gender, disability
Distributing seriously harmful misinformation is a serious offense
It is illegal to publish content that promotes self-harm
Cyber-flashing, which is the sending of unwelcome naked images to a third party, should be considered illegal
Also, flashing images should be sent to epilepsy patients with the intention of causing seizures.
Collins stated that these changes would "bring additional offences clearly within scope of the Online Safety Bill. Give Ofcom the power to make minimum safety standards for services they will regulate and take enforcement action against companies who don't comply."
Sentences in prison
This idea was recently promoted by Nadine Dorries, Digital Secretary. She warned of possible prison sentences for serious offenders and suggested that the two-year grace period should be reduced to three to six months. However, Ms Dorries' broad powers in the first draft of the bill should be limited, according the report. The draft bill's definitions of illegal content are too dependent on the discretionary powers of the secretary.
It was intended that Ms. Dorries and her successors would be able to exempt certain services, modify codes, give "guidance to Ofcom" and exercise powers on grounds of national security - which the committee believes should be variedly restricted, removed or subject to oversight.
Cyber-flashing is likely to be made illegal by the Dorries
Nadine Dorries' unlikely rise
While many of these changes were welcome by child protection advocates, including the NSPCC and others, some remain concerned about possible free speech issues.
Both the draft bill and this document provide exemptions for journalism and public interest as well as free speech.
The Adam Smith Institute (ASI), a think tank, said that the report did not address the "immensive threats" posed by the Online Safety Bill to freedoms of speech, privacy, and innovation.
The report recommends the removal of a controversial section on "legal but dangerous" content for adults. Critics had feared that this could lead to widespread unintended censorship.
Matthew Lesh, ASI's research chief, said that "The replacement - defining a series of ‘reasonable foreseeable risk' - is still worrying." It would still lead to speech being more restricted online than offline.
The report did not recommend banning end-to-end encryption. This has been criticized by politicians and child safety advocates for enabling criminal activity.
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It recommends encryption be included as a "risk factor", in the risk assessments tech companies must do under the bill.
The Internet Society, a non profit organisation that advocates for open internet, stated that the committee was "too eager to ignore" any attempt to compromise encryption.
Robin Wilton, from the society, stated that "the findings today are, unfortunately, a reflection a public debate heavily framed in misleading emotive terms of children safety."
"As a result, we see a bill which will lead to more complex and less secure online safety systems, exposing our lives for greater risk from criminals, hostile governments, and so on."
The government has now been given two months to respond on the report to the committee. The bill will be sent to Parliament, which is the next step in becoming law. This will take place early next year.