There are I think various models for legal liability of providers of websites like Fakebook.

Negligence is discussed above. I am not sure that the UK position would be the same and I am by NO means sure that an individual would have to go to the USA to sue Fakebook in tort if they were (plainly) in the UK and harmed in the UK. I doubt whether the traditional double actionability rule would exclude UK action if the harm was suffered in teh UK.

Contract may be another - the user terms require various things, and so if Fakebook fails to require those things of others they may have contractual liability to YOU if you are a member.

I believe however that Fakebook immunity under US law from defamation liability (and saying someone is a member of the BNP when they aren't, or is an old tart, etc is defamatory in my view) depends on so - called "safe harbor" legislation which requires the host to make available the identity of the abusive user. I should value a US lawyer's reading of that position.

In the UK, there is a recent judgment of Mr Justice Eady that I have not yet read in full but I believe is consistent.