The first of the two uses of to conquer
describes what the white horse with its rider did: they felled enemy after enemy, conquering territory after territory. That is consistent with the crown the rider had received before he left the stage. The second word, to conquer,
expresses his purpose, his goal. Even as he conquers the one territory, he moves on to keep conquering.
2 And I looked, and behold, a white horse! And its rider had a bow, and a crown was given to him, and he came out conquering, and to conquer.