Is 1GB RAM so little?
Psychological Explanation
Like with most things computer, it depends on the usage.
If I open only one browser tab, and that has a Youtube video in it, the playback will stutter. If I play something in Media Player and open a browser, the playback will stutter. A friend recommended to put more RAM in it, and indeed, Win7 system requirements state that 1GB is the minimum.
It sounds like the main issue here is a lingering, old-timey, outdated notion that web-browsing is a light-weight task that doesn’t require strong hardware. In the early days of the “web” (i.e., the mid-90’s), that was true and many people when buying a new system would sell their old one as a “web browsing system”. However, that was in the old days of simple, static HTML files. These days, web-browsing is indeed a power-hungry task because of all dynamic JavaScript, Silverlight, and especially Flash (which has always had extremely poor performance). The idea that old hardware can be used for browsing is simply no longer true; you do indeed need a (relatively) hefty machine to simply surf the web nowadays.
Hardware Upgrade
Intel Pentium 4, 3.2 GHz; 1GB RAM; Radeon X800 GTO
Those specs aren’t horrible (and they will certainly play many old games, including what used to be cutting-edge 3D shooters) just fine. However surprisingly enough, while it can play old 3D games with great performance, it may indeed struggle with just surfing “Web 2.0” sites (of which YouTube is certainly one), though simple, static sites should still be fine.
Obviously one solution is to add more RAM. Surprisingly, this can be a bit difficult because stores no longer sell older SDRAM and getting used hardware can be a chore because people often ask far too much for even old, slow, obsolete stuff (I blame shows like Antiques Roadshow for putting the idea that all old stuff is valuable in people’s minds).
OS Optimization
As others have mentioned, Windows 7 is itself rather hardware-hungry and will gobble up (read waste) quite a bit of memory, especially with a default installation. If possible consider using XP or Linux or something lighter. If not, then you can lighten Windows 7 to make it less burdensome on the hardware.
To optimize your (or anyone else’s) old system for surfing or any other task, you should minimize Windows as much as possible. There are plenty of good guides that help you decide what to cut, but Black Viper’s Windows Service list is usually a good place to start. Turn off any and all unneeded Windows services, then use something like Autoruns to disable any and all unneeded programs from automatically running.
Once you have stripped Windows down to the basics, then you should have a lot more free memory (and CPU cycles for that matter) available for your designated task(s).
Drivers for Performance
Of course you’ll also want to use the latest drivers for your hardware which may be a bit difficult to determine/find for anything that is discontinued like your X800 for which Catalyst 10.2 was the last driver released.
Some Browsers are More Equal than Others
Another factor that affects web-surfing performance is the browser. For example, when the very first version of Chrome came out, it was blazingly and shockingly small, light, and fast. Over time, like most software, it became bloated and slow. Try different browsers to see which gives eh best performance on your system because they are definitely not all equal in this respect.
Lightweight YouTube
Finally, in regards to YouTube in particular, try switching to the HTML5 player to see if it performs better than the Flash player as well as the Feather version. Also try YouTube Center to simplify and tweak YouTube.