Imac g4 won't start but shuts down when apple logo appears on screen

Are you sure the iMac is really shutting down or just turning off the backlight? The tell-tale sing would be to let the unit fully boot-up (wait 3 minutes) and the press the button the apple keyboard that ejects the CD-ROM. If the CD-ROM door opens, you iMac is running with disabled or broken backlight.
The inverter circuit on the iMac G4 is known to be weak.
There could be another problem, though. The iMac G4 is also known to have weak internal power supplies. Here's how you can tell:
Hold down the CMD OPT "O" and "F" keys at the same time while pressing the power button. This will tell the iMac that you want to boot into the Open Firmware shell, bypassing the Mac OS X startup. If this works, your power supply is good, the inverter is good and you have a problem with your Open Firmware boot string. To fix the boot string, Power on the unit while holding down the keys CMD-OPT-P-R to zap the parameter RAM (PRAM) and reset everything to defaults.
If you were not able to boot into the Open Firmware shell, your problem is either a failed power supply or a failed screen backlight. The last possible culprit is your iMac could suffer from "Bulging Capacitor Syndrome", where a batch of iMac G4s, iMac G5s another electronics were fitted with a bad batch of capacitors. Eventually, the caps will rupture and cause irreparable damage to the logic board. Only way to verify this is to pull the unit apart and visually inspect. If you see little green or blue cylinders that seem to be bulging or oozing a thick oily goo, that's your problem. Only recourse is to shop around on eBay and look for a replacement logic board.