The baby seat is a great tool: the baby can sit comfortably, can see everything, and can be rocked easily. That’s fine but it weighs 5 kg at least without a baby. With a child, it can weigh even 13-15 kilos.
Carrying the seat has also its technique. Try to carry such weight in front of you and not on your side.
If you swing it on your side you must keep it a little away from you, so the weight is increased significantly.
Think about it: 13-15 kg, holded away from your body on one side. This means a pulling force on your spine and muscles that they cannot tolerate too long. If you carry the seat in front of you holding its handle, leaned against your hip, it will put a lot less weight on your spine. Your bicepses will work quite hard that’s true but you can avoid the pain in your waist and scapula.