'Before' would not be appropriate, because it would refer to a time period with known beginning and end, both before, not after or during, a specific point in time. In the sentence the idea is rather different: the person turned the bandage repeatedly without knowing how many times he or she would have to keep doing it. What he or she did know was that stopping to turn the bandage would happen at the same time as another thing: the bleeding stop. So with BEFORE you mean X begins and ends before Y; with UNTIL you mean X begins before Y but ends together with Y.

No, that wouldn't work. The "until it stopped most of the bleeding" continues the thought of turning the bandage. "Careful not to hit the bone" is a clause that interrupts the flow of the sentence, but "until" isn't referring to it. The action being described is a tourniquet: you keep turning until the blood stops flowing. What you're proposing is: "She turned the bandage... before stopping the bleeding", which isn't what's happening, There's a cause and effect of turning the bandage that causes the bleeding to stop. Using "before" would indicate that there's two separate (independent) actions with no correlation of cause and effect: (1) she turned the bandage, then (2) stopped the bleeding (the "how" or "with what" is unknown).