Thanks for all the replies so far, the general consensus seems to be that it could be an IE6 problem. The problem is, I don't know which one to try & install, mabmac mentions both IE9 and IE11.
As a not too technical user, I've never actually upgraded my browser before, so should be interesting....
Mixture, the banking site lets me into the main website as you say, but then displays the can't display page message when I try to log in. My main email site, (hotmail), won't even let me access the main page - it displays message as soon as I type the address in the browser & hit the enter key. Confusingly, it has no problems letting me in to my gmail account (only used as secondary email).
Must admit, it's hit and miss which sites it will let me on, & I'm completely baffled as to why it is acting as it is!