If the car is turning over then it probably is not the battery nor the alternator, because the battery is getting the power to crank and eventually turn over. Problems like this could be any of a group of items that is causing your car to be sluggish turning over. Could be low on oil, you could need a new fuel pump, fuel filter, spark plugs, etc... Fortunately if it is any of these that is causing it, then they are simple and inexpensive solutions.