I have never seen a horse stop growing at 2 yrs old... Their hock growth plates don't close until 4-4.5 yrs old, with all of the growth plates closed at around 5-6 yrs old.
My mare grew a full inch from 4 to 5, then filled out (QH/Appy, Impressive grandaughter). She ended up 15.2h, and she was about 14.2 at 3yrs old.
My husband's ArabxTB grew almost 1" after he truned 5! He ended up 15.2h as well. He never did fill out much, but he is half Thoroughbred, lol.
Our new Haflinger gelding will be 5 in March and he's grown about 1/2" since we got him last August.
I think it's more normal for horses to grow slowly. It's the horses that get fed tons of high protein feed early on that grow quickly in height, and then often end up with arthritis or other issues earlier than normal. A horse should not be arthritic or need joint supplements/injections at 10-13 yrs old...