That sounds like a Late 2004 iBook, but a 14-inch display would be on a 1.33 GHz. There is no iBook with a 1.6 GHz processor.
Boot to your 10.4 installer again. Open Disk Utility. Choose the hard drive, which would be the top line, showing the manufacturer's info with model number, not the line with the drive label. You will see a Partition tab. In Partitions, choose the drop down, and select 1 partition. Choose the Options button, and make sure that Apple Partition Map (APM) is selected, and then make sure that Mac OS Extended (Journaled) is the partition type, then name the partition (whatever name you like) and press the Continue button. That should absolutely be the best steps to make sure the drive is ready to install OS X. It is also very possible that if you have the original hard drive, that may be failing. It's 15 years old.
When the Disk Utility, with a new single partition, is done (should only take a few seconds, assuming the drive is still OK), then choose Shutdown from the menu. After power is off, restart to the 10.4 installer, and try to install the system again. If it fails (again), you should get some kind of error about that, even something like "some software could not be installed", or something like that. Need to know the exact message, or even that you see no message at all, and you get a kernel panic (a screen that may say you need to shutdown your computer, in multiple languages)
