Q: The car turns over but doesn't actually starts

An engine is stalling inspection will reveal the cause. When the vehicle is cold, the mixture has to be enriched in order for the engine to run initially. That enrichment signal is sent (to the fuel injectors) if the appropriate temperature sensor outputs are received at the PCM. There are other controls such as an idle air control valve that would have to be checked. It is possible you have a large vacuum leak, including that from a stuck open EGR valve, which on a cold start really throws off the air fuel ratio. It is possible the fuel pump relay is cutting out. Other possibilities include a failing crankshaft position sensor as well as problems with the ignition coil(s). There are additional possibilities and consequently the most efficient thing to do is schedule a mobile engine stall diagnostic during which a YourMechanic professional will quickly identify the root cause and then the repair can be implemented right away, too.