Like R.W. Burgess said, have you checked the battery voltage with the engine running using a volt meter (VOM or a DVOM)? If it is 12 volts or less car not charging. Check the voltage from the housing of the alternator to the positive alternator terminal as one check, and then check the voltage at the + & - of the battery. If you have about less than 13.5-14 volts you have a bad alternator, if you have that voltages it would indicate the car is charging so you have other problems. As for the car turning off, that is very possible. If the battery voltage on most 12 volt cars drops below about 9.6 volts, the electronics of the car starts to shut off. Not quite as critical for cars prior to electronic ignition, but is true since.