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).