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If I'm turning left into a supermarket car park and a driver facing me is turning right into the car park, which rule applies - the left turn v right turn rule (in which case I would have right of way) or the major road v minor road rule (in which case they would have right of way)?

Because you are both on the same road, the left turn vs. right turn rule applies, that is the right turning car should give way to you (turning left).