Ok, that sounds like a browser then as you are referring to being at websites.
I don't have an answer as such but it seems that clicking on your desktop shortcut is trying to open two websites at once (may be the same one twice). Do you let it finish to see what comes up on the tabs ?
The other thing I can suggest is to right click on the shortcut that you are left clicking on , selecting properties, and scanning what is there to see if it looks as if the action in clicking is to open two webpages. If so it would have to affect all your shortcuts, which sounds strange, unless your PC has been infected. You could always try an AV scan and a malware scan.
Or it could be Windows corruption. I'm sure someone else can suggest how one checks for that: I know you can get Windows to check itself I just can't recall how.