TenFourFox is a Firefox updated alternative for older PowerPC based Macs.
I don't think you are going to have a more responsive computer installing Leopard, but you can surely free up disk space, moving some of less used data onto an external hard drive.
If the internal disk is almost full, that can impact on the performance because there will be not enough space for swap files.
You iBook has a built in 512 meg of memory. So from what you have spec you have another 512 meg installed under the keyboard. My suggestion would be to remove that and replace it with a 1GB of DDR memory.
That will take your iBook up to it's max of 1.5GB memory. That extra 512 meg will help you speed it up. Memory is a cheep upgrade and even more cheaper for older memory modules.
Clear down any files to a USB or FireWire external drive, or install a larder hard drive. Maybe a 150 or 200 GB. Extra space is always helpful.
When it comes to the Internet and your browser, really isn't much you can do. There is little support for Mac OS X 10.4 Tiger when it comes to vast improvements to coupe with modern web sites.
Like Alberto has already suggested, you could try TenFour Fox web browser. I have to say from experience it's not perfect in performance, but maybe better supported than FireFox for Tiger.
Best of luck.
You may need to start a thread of you own and include the technical information about your iBook G4 such as the build year, hardware specifications, total capacity of the iBook's internal hard disk drive, installed RAM total, and if you know it, the available free-space remaining in the hard disk drive. Disk Utility could help, as could Activity Monitor, for finding information on the system hardware details.
If the computer has all the RAM it can use (according to MacTracker.ca or other spec source) and a large enough capacity of unused free space in the hard disk drive, it could be capable of running Leopard 10.5.8; but you would have to get a retail install disc through Apple online or by phone through Support since it probably is not available in their store. A call to the main Apple support may be worth a try should you want to upgrade the OSX to 10.5.8 if the computer you have is able to adequately run this version.
Interest in these older notebook portable and pre-Intel based Macs has slowed, so replies may be long in arriving. Other forums have disappeared over the past five to eight years, too; those had good exposure.
Hopefully this helps somewhat.
Good luck & happy computing!