Twitter and Facebook should let users block anonymous messages if they are serious about stopping bullying and trolling on social media sites, David Cameron's adviser on child internet safety has said.

Claire Perry, a Conservative whip, said internet firms are currently not doing enough to tackle bullying online and called for more prosecutions of people who make online threats,that she described as misogynistic.

In a hearing with the House of Commons media committee, she said bullying would be "driven down" if users could choose to block communication from anonymous users.

Perry, who received online threats over the summer, said there should be an online verification process, so people can see if they are dealing with other users who have supplied their real names or chosen to remain anonymous.

"Having been on the receiving end of a storm of Twitter abuse, I don't think the companies do enough. Part of the problem is anonymity of usage," she said.

"People post about how they'd like to rape you and kill you because they think you don't know who you are. If there was some way of the company knowing and being prepared to verify that identify and to show you that verification, I think it would lead to a diminuation in that kind of behaviour.

"I don't think the companies do enough and I think there is a great concern around it given the US legal framework around which the global companies operate."

Perry is the prime minister's adviser on the commercialisation and sexualisation of children, concentrating on the impact of online pornography, but she is also working on the issue of internet bullying.

Asked whether there should be more prosecutions for trolling, she said: "Yes I do. I think it is deeply misogynistic. We did see cautions and apologies [over the summer]. But I think actually prosecucuting for what I think is pretty vile behaviour would be helpful.

Ed Vaizey, a Tory culture minister, said he thought there was "mileage in sitting down with the social media companies" to discuss Perry's suggestions.

Earlier in the hearing, a Twitter executive said anybody could use an alias, while Facebook said it asked people to use their real names but did not carry out checks unless there were complaints.