There should be no pain whatsoever after a shot, properly administered.
You state you are injecting "in the side".
Lantus injections can be given in the fat of the belly, at a slight angle. With a new sharp fine needle, and fresh Lantus, the discomfort should be insignificant.
Talk to your health care provider about needle technique. Usually the nurses know more about this than the physicians.
I just remembered, someone said that to reduce stinging, try releasing the skin you've pinched before you inject. I don't know if that works or not. For me, it's my Apidra that stings sometimes, though it isn't known for that.
Is it Lantus? That one is known for stinging, though I don't have the problem with it. Only ideas I have is to try a different site and see if it helps. The belly is actually the best site for absorption, but not sure if it will help the stinging.