I think you'll find that there is no definite answer to this, as each horse is different and each can have (or not) growth spurts at any age between ''young'' and say 4. But, I have always been under the impression, they have reached their adult height by 5, and matured by 7. I'm sure people may say different things though.
Saying that, some say add two hands for a yearling, one hand for a 2 year old.
There are various ways of estimating how big they will grow.
By the age of 6 for a mare and 7 for a gelding they are usually fully grown although natives and large horses (17hh+) usually carry on the longest.
From 18 months on you can do the string test.
Take a length of string and measure from the point of the elbow to the centre back of the fetlock. Reverse the string so it is from the point of the elbow to over the withers. The difference in the length is how much they have yet to grow.
Not totally accurate but near enough.