What makes a new baby with facial bruising get jaundiced?

Jaundice. I agree with that answer but having bruising from birth trauma even natural birth trauma that can occur, does increase the risk of elevating the bilirubin in the blood thus causing jaundice.
Bilirubin. Newborn jaundice occurs when a baby has high levels of bilirubin in the blood. Bilirubin is a yellow substance that the body creates when it replaces old red blood cells. The liver helps break down the substance so it can be removed from the body. The facial bruising is unrelated; that usually happens when the baby's head moves through the birth canal.
Same as without. At the end of pregnancy, the ongoing change over from fetal to air breather blood cells causes an increase of bilirubin waste that is disposed of by the mother's system. Once the cord is clamped at birth, the work must be done by the baby. The newborn liver takes days to catch up with the work. Jaundice is just bilirubin backup, from any source, including bruising. It fades when they catch up.