Well it is true that many a website are not supported on those older web browsers. Flash has been updated many times, and Java on 10.3 or earlier is barely supported on most Java websites. Javascript is less troublesome, That said, if you installed 10.3.9 combo and the Java update for 10.3.9,you can follow the tip for web browsing below:
http://support.apple.com/kb/DL496
http://support.apple.com/downloads/Java_Update_for_Mac_OS_X_10_3_9
https://discussions.apple.com/thread/4781929
And find which web browser has the best 10.3 support.
Any website that does not follow w3.org standards or use server side processing is likely to have trouble with your machine. You can certainly write to the webmaster and inform them of http://www.anybrowser.org accessibility guide if a website needs accessibility to the older machine.
That said I've also written this tip of all the major web browsers for the Mac*:
http://www.macmaps.com/browser.html
If you find the thread above is insufficient.
If you can tell us the website you are trying to connect to, we can test and see if it is not following enough standards for you to connect or you actually have a network connection issue.
* Links ot my pages may give me compensation.