It's not so much websites are moving away from IE. It's more that Web standards dictate the use of CSS. CSS is supposed to mean that pages are viewed exactly the same despite whatever browser you use. Style sheets have been around for about 10yrs now, yet different browsers still don't render the code the same as they are supposed to
IE is one of those browsers that had a reputation for not being able to interpret & render CSS properly, but there are still some others that have quirks. For example see the screen cap from bcat at the top of the forum thread showing how a particular browser isn't lining up a graphic how it should. I spent hours doing the css and about another hour last night checking and double checking my code and seeing if I could tweak something but still no joy so far. It's very frustrating for web designers when browsers behave unexpectedly so I can fully understand why some web devs give up if IE doesn't render a page how its supposed to.
Re browsers: - Netscape was the very first browser that I used as it came on a floppy disk with the welcome pack from my first ISP. MS IE wasn't really heard of back then and until MS started packaging it with their operating system (think Win98?) no one hardly used IE.
I remember Netscape Gold being released and thinking it was the bees knees when they brought that particular version out because of the suite packaged items such as navigator, email client, news reader and what must have been one of the first freebie WYSIWYG html editors. I can't recall the year exactly Netscape Gold version being released but possibly 97? It wiped the floor with IE which in the late 90s wasn't very good and quite basic compared to Netscape.
I stuck with Netscape for many many years, only moving to IE when AOL bought Netscape out and ruined it by making it bulky and slow, just like AOL ruined ICQ which was the best messenger client ever. AOL killed both
iirc Mozilla came to the rescue of Netscape - Firefox is built on and based from Netscape code.