Problem with car starting up since timing belt and water pump were replaced.

What is your question?

Last week my car broke down on the parkway. I found out the water pump had broke. Had timing belt and water pump replaced. Since then, having a problem when starting the car. It's ok when first starting for the day, but once I am driving and then make a stop, when I turn on the car again it starts for a few seconds, then cuts out, then goes back up again. Also when driving 50 mph or more it stalls a bit sometimes, but not all the time. I went back to the mechanic who had replaced the water pump and timing belt. He said the air temperature sensor was broken, and it might have given mixed signals to the engine. He replaced the air temperature sensor, but the problem with starting the car is still there. Do you have any ideas on fixing this problem? I am not confident about the mechanic I'm using. I paid an extra $80 for the air temperature sensor and it hasn't fixed anything. Thank you for any answers you can provide.