When I turn on my camera it shuts automatically.

Camera specifications for the batteries required by this unit are:

Kodak NiMH battery pack KAA2HR, or 2x AA

If battery pack is installed try using a couple of regular AA size batteries to make sure it is not just a charging problem. If camera works with regular batteries, then battery pack and/or charger could need replacing.

If camera still not working after testing, a visual inspection of the camera's interior components may be of use. A teardown tutorial can be accessed at this location:

Kodak Easyshare z650 Teardown