I notice before and after I installed my supercharger, my car always had throttle lag and slow acceleration until it warms up....is this the norm?

 

Why are you diving the car cold, and No it shouldn't be but you also should be slamming the gas when the motor is cold :/

 

Who tuned it? There should be a table in the software that governs warm up enrichment, where it has you running richer when the coolant is at certain temps. Ask your tuner about it. He'll probably ask you to log a cold start. With a cold start log, it's easy enough to fix in those tables.