We introduce a structure on a Garside group that we call Dehornoy structure and we show that an iteration of such a structure leads to a left-order on the group. We define two conditions on a Garside group G and we show that if G satisfies these two conditions, then G has a Dehornoy structure. Then, we show that the Artin groups of type A and of type I2(m), m≥4 satisfy these conditions, and therefore have Dehornoy structures. As indicated by the terminology, one of the orders obtained by this method on the Artin groups of type A coincides with the Dehornoy order.