Searching based on members that have a twitter website address
When you say "return all groups" I presume you mean return all contacts, as I don't see a way to associate a website with a group, but please clarify. From what I can see, there doesn't seem to be an option to search on website at all in Advanced search, but I think you can achieve what you want in Search Builder with

record type: contact
field: website
operator: like (or maybe a more precise regex)
value: twitter.com