Chim,
Have you ever thought of ditching Opera and use Firefox?
I even received an update to Firefox version 15 the other day, and now have version 2.16, the update was for security reasons.
It works fine on Windows 98, the reason why I mention it to you.
You may even find that your web problems are fixed when using Firefox, by all means give it a try.
If you are interested in Firefox then download version 2 only as version 3, well guess what, doesn't work on 98.
Later tonight I will test these sites you have problems with, can you post the links to them here, so I will have access to the exact pages you yourself have a problem with, and I will let you know how I get on, being that I use Firefox.

Heh Heh Heh! I actually tried
Firefox 2.0.0.14 for about 2 weeks BEFORE I ever tried Opera. It had potential for a while. I really thought I WAS gonna have a use for it, that it WOULD have its niche. But, real quickly, it fell out of favor with me and wound up in my Browser Doghouse.

It had 2 extremely annoying habits. It had a HUGE problem of NOT opening up Links / Pages on the 1st Shot. It would start loading and after 3 seconds or slightly more, it would declare the task done when it was obviously BS and there was nothing there. Then I'd try clicking again and it just wouldn't take. It kept indicating - DONE! I'd have to try multiple attempts at opening the Links / Pages on New Tab after New Tab ... New Window after New Window. I'd Close and Reopen Firefox. And / or I'd get nuclearly Peeved

and rapidly click about 10 to 15 times on the Link by the time Firefox would decide to get out of Browser Couch Potato mode and open up the page finally.
The other Bad Habit was that after it already had me FUMING from the stunt above ... it then had the unmitigated gall to NOT display the pages in their entirety. Sometimes it would take multiple Reloads before the pages were finally complete.
So, I remembered vaguely having seen
Opera mentioned here and there in my Net Browsing sessions of Reviews of Software, Anti-Malware and such. I Googled it and liked what I read. I downloaded it and tried it. My first 20 minutes with Opera were NOT very auspicious and I was probably seconds away from aborting that experiment. But, luckily I took a break and afterwards things started to click. Opera 9.27 was virtually perfect. It easily neutralized all the aforementioned Firefox inadequacies. Needless to say --- I Ditched / Kicked Firefox to the Curb. I still have it on my Desktop. It's just benched ... 2nd String Quarterback with the Clipboard.

It is really Opera 9.50 and 9.51 with which I am having to exercise some patience and tolerance ... hopefully temporarily. But, even with those 2 versions' Bugs, they're still better than Firefox ever was on MY Quirky system. Nah! I would never think about ditching Opera just because the avast! Web Shield doesn't work properly with it on MY computer. I've never had Web Shield functionality, so I can't miss it.
As for the brutal
WikiWeb Test that instills FEAR into Browsers? Here are the URLs:
http://en.wikipedia.org/wiki/Main_Pagehttp://en.wiktionary.org/wiki/Wiktionary:Main_Pagehttp://www.merriam-webster.com/On the Wikipedia and Wiktionary, pay close attention at the bottom where it has these "Wikipedia's sister projects" and "Other Wikimedia Projects" Icons. With the Web Shield enabled, those are real easily the clincher. They're always missing then. But, other content can also be messed up or missing.
On the Webster site, that one is just basically usually some obvious, can't miss it - MESS.
But, it's really NOT just those sites that fail with the Web Shield enabled. It's downright virtually almost ALL sites. Those in my WikiWeb Test just happen to be the first 3 that I try as they're the first 3 on my Opera Personal Bar. The WikiWeb Test is no doubt going to PASS on YOUR computer as it no doubt will PASS on everyone else's computers. It's probably just a syndrome isolated to
MY OLD Slow computer and
AOSnaiL Dial Up connection.