My daughter is now going on 18, but she is still only 4-10". I've been modifying guns for her since she was 6.
There are youth rifles available now that weren't around 10 years ago, but they don't make anything in a youth in a .30 cal. They also haven't learned about keeping the barrel shorter so that the front end weight is reduced.
My daughter has, and shoots, everything up to and including .375 HH and 458 WM! Everything has had the stock shortened to 12.25, and the barrel shortened to 18.5 - 20" depending on the calibre. Her primary 2 hunting rifles are a Stevens 200 in .308 (18.5") and a Savage16 in 300 wsm (also 18.5"). So far she's shot her share of deer.
Find a rifle that starts out fairly light, ie a synthetic stock, and then modify it for your own needs. Remember, if you are 6' 200lbs and find a 8 lb rifle is hard to hold steady, or carry all day, imagine if you were 5' and 110 lbs!
Start light. Reduce the LOP and the muzzle weight. If in doubt, put a limbsaver or decelerator pad on the rifle, but generally smaller, lightweight people have very little problem with recoil as they tend to roll with it, rather than absorb it. My kid has been shooting centrefire since she was 8, and shot my 450 marlin the first time off the bench when she was 9 or 10! Recoil really doesn't phase her.
The Ruger compacts and frontiers are excellent rifles, and I actually own a couple of them, but they are pricey and really not all that light. The compacts in a blued walnut are the lightest, but if you get a stainless laminate, or a frontier, they weigh 7.5-8 lbs!
I would recommend the Stevens 200 at under $400 with taxes. Chop the barrel, another $50. Shorten the stock, $35-75 depending on if you add a better pad. Trigger job another $50-75. $575 before a scope and you have exactly what you need! A lot less than a compact at $750 +++