What is a personal injury claim actually made of when the other insurer disputes fault?

Someone ran a light and hit me, I've got injuries and a wrecked car, and now their insurer is suddenly acting like fault is unclear, which it absolutely isn't. I thought a personal injury claim was basically just "you hurt me, pay up," but it's clearly more structured than that and I don't understand the pieces. Is the claim the medical bills, the car, the lost work, all of it bundled, and does disputing fault freeze the injury side or are those separate tracks. Trying to understand what I'm actually putting together here before I either deal with their adjuster myself or decide it's over my head.