Of the 255 Scottish adults polled for child protection charity the NSPCC, 9 per cent supported statutory regulation, while 63 per cent disagreed that networks do enough to protect children from sexual grooming.
It comes as part of a wider survey that found nine in ten adults across the whole of Britain also supported the idea of regulation.
The charity’s latest campaign, Taming The Wild West Web, is calling for the introduction of a “robust independent regulator” to force social networks to protect children using their platforms.
Ruth Moss, of Edinburgh, whose daughter Sophie took her own life at the age of 13, is backing the campaign.
Ms Moss described how it felt after finding out her teenager had looked at self-harm and suicide content on social media.
She said: “Sophie’s death devastated me. No mother, or family, should have to go through that. It was so unnecessary, she had so much to live for. She was only 13.
“I found out that she had been looking at completely inappropriate things online. Some of the images were so graphic.
“She was also communicating with people in their 30s and pretending to be older than she was, under a made-up persona.
“Whilst the internet was heavily controlled at home and at school, Sophie had free wifi when she was out, making it very hard to ‘police’ her internet use 24 hours a day.”
Ms Moss added: “Social networks should have a duty of care to protect children and vulnerable people from damaging material and self-regulation is clearly not working.
“The protection of our children is too important to leave to the goodwill of large, profit-orientated organisations. Statutory regulation is needed and as a matter of urgency.”
The proposed regulator would have legal powers to investigate firms, which would require to meet a set of minimum standards, and impose tough sanctions for failures including fines of up to £17.5 million.
Peter Wanless, NSPCC chief executive, said: “The support for statutory regulation of social networks is now overwhelming. It is clear that society will no longer tolerate a free for all under which tech firms allow children to operate in a precarious online world with a myriad of preventable risks.”