Answer:
A great way to improve your turnout is strengthening your legs. Once you have enough strength and learn how to use your muscles, everything clicks. You will engage your muscles easily and your turnout will come naturally. Stretches may help, but you need the strength to really improve your turnout forever.
Be careful when trying to improve turn-out. Each individual has certain limitations in their hip socket, so forcing turn-out can lead to major hip/leg problems. Always go slow when stretching, and don't stretch too hard when you are cold. Below is actually called a 'roll through', which when done correctly is very good for dancers. It is extreme though, so again be careful.
Sit with your legs in a wide side split. Stretch to both legs and the front. Put your hands in front of you and then walk yourself over until you are laying in on your stomach, with your legs in the same position. Lay here for a bit, then bend your legs at the knees, folding your feet in towards each other. (You'll look a bit like a frog.) Do this each day and it will greatly help your turnout.
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Keep your knees up while in the splits.
My ballet instructor recently taught me a good exercise. You stand against a wall with a chair in front of you for balance and you turn out your feet.
The frog position is also good for turnout, but if you hold the position for too long you could really mess up your hips. you just lay on your stomach and put your feet together with your knees bent and as far apart as possible. let the weight of your feet pull your feet down. you could also have someone else push down on your feet, but too much pressure is very painful.