That's because felt recoil is a result of the weight of the gun divided by the mass of shot being accelerated. Small, lightweight guns are actually MORE punishing to young shooters (and yourself, as you discovered) than full sized guns.
But, you may say, my son is too small to handle a full sized shotgun. Not to worry, if you look at the formula above, you'll see that there are several ways to mitigate the problem.
First - the amount of shot. All else being equal, a 1 1/4 ounce load will kick much harder than a 7/8 ounce load. Buy the lightest loads you can find (Don't worry about the number of pellets in the shell. Good shooting depends on the shooter's ability and the density of the pattern at the target distance, not the amount of lead in the air. Try to have the right choke for the distance, or adjust the distance to the choke you have.)
Secondly - the weight of the gun. All else being equal, a sub-gauge gun will always kick harder than a full scale 12 ga. So, what you can do is to add weight to the gun. The best scenario is to add weight equally to both ends of the gun in order to maintain proper balance. But, to do so in this case would make the gun just as heavy as a full sized gun and difficult for your son to manage. This is a rare instance where adding weight to the gun is more important than maintaining balance. In my opinion, the weight should be added to the butt stock, where it will be well supported and not make swinging the gun more difficult for him. Filling the through-bolt hole with a small sack of lead pellets would work, for example. Even a side-saddle shell holder on the butt stock can provide a little relief. The loss of balance won't mess up his shooting as much as the sharp recoil does.
As you probably already know, once one's shoulder gets beat up, every subsequent shot can be painful. Conversely, though, if you avoid bruising the shoulder, shots that would be painful to a bruised shoulder won't be a problem. The idea, then, is to start slow and gradually increase the numbers of shots taken at one outing. It might be hard at first to quit shooting after only one box of shells but, if you can be disciplined in this way, his shoulder will become tougher without getting to the bruised stage, and he'll be able to handle hours of shooting at a go later on when his shoulder is accustomed to the workout. (Depending on the recoil, even just one full box can be too much shooting at first.) ETA: Adequately padding his shoulder (without adding too much bulk) will help keep it from bruising easily, too.
Making target shooting easy at first and gradually adding difficulty can build confidence and, once hooked, he'll never look back. Consider throwing only straight-aways at first, then add angles and more speed, if you have the option. Most of all, make it fun. My son got beat up by the gun when he was starting out. He hasn't shot for years. My daughter got beat up at first, too. But she's a different personality - she just gets tougher in the face of adversity. Last year she won the Alberta "B" Class championship in International Trap.