the biggest problem is browser sniffing, i.e many websites (specially msn sites), check if the browser is Opera and if it is then they simply send bad code to Opera. Or they check if its not IE, and then send broken code.
TO completely hide opera do this:
1.right click > site prefs > network tab > mask as internet explorer .
2.right click > site prefs > cookies > delete all cookies manually
3.goto tools > perf > advanced > history > clear cached
open the site
If it still doesn't work ( very very less chance)
1.disable javascript
2.repeat the above steps
open the site
After doing this practically ALL sites will work in Opera flawlessly, atleast all that open in firefox .
Obviously sites designed to 'work' around Exploder's bugs still wouldn't. I haven't encountered a single site that doesn't work in Opera since I came to know of the above method.
For more info:
http://my.opera.com/community/openweb/
http://my.opera.com/community/forums/topic.dml?id=4893
MAKE SURE TO GOTO > HELP > REPORT SITE PROBLEM, WHENEVER YOU ENCOUNTER ANY DUMB SITE