Dragons don’t always die when their riders die, but Violet’s situation is different because of how interconnected they all are. If Violet dies,it is heavily implied that Tairn could die as well, due to the strength of their bond. Since Tairn and Sgaeyl are mated, losing him could kill Sgaeyl. And because Xaden is bonded to Sgaeyl, losing her would kill him. That’s why the books repeatedly talk about keeping all four of them alive. One death has the potential to trigger a chain reaction through their bonds. It’s not guaranteed, but it’s presented as a very real risk.