I recently purchased a quarter pony for my daughter. This mare has minimal training but is sweet and willing. She seems a bit stiff to me and I am not sure if this is the QH in her or if there is something we can do to make her a little more supple.
Horses do not become supple on their own. They have to be trained to be light responsive and supple. The Quarter Horse in her has nothing to do with it. They still have to be taught suppleness. I could write a book on this but obviously do not have time here. You might just bend her around with a snaffle bit to achieve this. Some horses need to be bent around more that others, to be kept light and responsive.