Well the Links folder is just put there by IE not long after it is removed. Sorry we can't help get rid of it, because that's how its designed.
Either put up with the folder, or look for an alternative. There are numerous other browsers out there which give IE a run for its money, such as Netscape/Mozilla, the standalone Mozilla Firebird, Safari, ummm, those stand out as very high quality browsers to me. Did I miss any?