I bought My son a Winchester Mod. 70 Jr. model when He was 11, and He killed His first buck with it. It was a dandy gun to start Him off with, but He was a big kid for His age. He had no problem handling the kick, even out of that lightweight rifle. I roll My own, and lightened up the powder in His shells. But I think most young'uns at 13 years of age can handle the kick of a .308.
I'm with PAJack on recommending the Rem. model 7. When My son outgrew the Model 70 Jr., I bought Him a model 7. He loves it. My next rifle is going to be a Model 7.