You aren't worried for nothing, a LOT of seats simply don't provide enough structural support for the seat to fit snugly with the normal seatbelt whilst the vehicle is moving. 

If clips don't come with the carseat, you should definitely look into extra safety clips, see the final tip from my recent article on this very subject: 'Choosing A Child Car Safety Seat: 4 Tips Every Parent Should Know'- there's a product called SeatSnug that can provide the extra safety these seats need which solves the problem immediately, but makes sure that the kid is as comfortable as normal.  Definitely look into getting that extra bit of kit for the car journeys you make!