Over here (Finland), the mandatory insurance covers passengers (and driver) also in your car, not just third parties. Different countries obviously have different practices. I can see that this is a problem in countries where the vehicle's insurance does not cover its own passengers.
(In fact, the mandatory car insurance here even covers personal damages of third parties at fault if they are pedestrians or are riding a bicycle; i.e. if a bicyclist runs a red light and collides with a car and hurts himself, the care cost is carried by the car's insurance. If the bicyclist hits a tree, the tree has no insurance... this applies to personal injuries only, not the broken bike. Any damage to the car is covered by the bicyclist's insurance if any.)