If it turns over at a reasonable speed there's no issue with battery or starter motor - the issue has to be immob, fuel (cleanliness, pump, injectors, ECU), ignition (ECU/sensors/coils/leads/plugs) or compression (HG, valves, rings).
If it turns too slowly to fire, the issue is somewhere in the chain of battery, wiring to starter motor (including earth the engine/transmission), or starter motor.
It's nothing to do with the alternator. That's just to charge the battery and power all electrics when the engine is running.
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