At the bench the perceived recoil is much greater.You are trying hard to not move, to be steady, to get a good hold and unfortunately for many they anticipate the rifle going off.This creates the flinch from expecting the recoil.You upper body is going to feel much more force when seated as opposed to standing, also when shooting at game you are focused on the animal more and do not notice the recoil, sometimes adrenaline pumping causes people to not even remember squeezing the trigger or feeling the shot.
Purchasing a Caldwell leadsled would be a great tool for your sighting in and practice time at the range, it absorbs a huge amount of recoil and steadies the rifle for sighting in.
Practicing with a small chambering like others have suggested will get your technique figured out so that you can be more comfortable shooting and develop positive skills.
I do also think that deciding on a small chambering is not the solution to your issue but practice, fit of rifle and technique are.
You may want a rifle that allows you to practice lots without punishing your shoulder and allow you to grow confident in your ability.
So, IMO a good practice/target/hunting chambering would be a modestly weighted rifle with a scope in something like .260, 7mm-08, 257 , 25-06 , 308, or similar.
Remember that comfort and fit are the biggest factors when developing good shooting skills and accurate groups.
Hope this helps.