This issue is odd - looks like your profile is somehow preserved (maybe for some odd reason there are insufficient rights to modify the profile folder).
Could you please do the following:
go to opera:about
check the paths where your cache and profile is stored (assuming you are on Windows, these would probably: C:\Users{user}\AppData\Local\Opera Software\Opera Stable and C:\Users{user}\AppData\Roaming\Opera Software\Opera Stable respectively; I'm not sure about Mac or Linux)
close Opera
go to the folders - you can remove the whole "Opera Stable" cache folder but if you want to have your profile preserved for future use, just rename "Opera Stable" profile to anything you want
restart Opera
Your Opera should now restart with a completely clean profile. Please browse a while, close, restart again and check if it solved your issue.
