I will not comment on the use of "curls" at all. Here is how my PH mentor, the late Jan Marais, taught me:
If you draw a line from the start of the horn more or less in the direction of where the start points to, you will see that the line is pointing AWAY from the center-line of the kudu's head. In the photo posted by the OP, just above his rectangle marked "1st Turn" the ridge along the spiral disappears from view. About this is the start of the first turn, because if you just go a bit higher, and again draw a line more or less in the direction of where the horn is pointing, this line now points INWARD, i.e. towards the center-line of the kudu head. The direction in which the horn points has changed from first pointing OUTWARDS to now pointing INWARDS - this change of direction is the " 1st Turn" on the horn.
Again with reference to the kudu picture posted by the OP: Just above the last bit of green of the tree directly behind the left horn the DIRECTION to which the horn is pointing changes to the OUTSIDE of the center-line of the kudu's head. This is the 2nd turn.
Again with reference to the kudu picture posted by the OP: Just below the point where the first very thin twigs of the tree on the left "disappears" behind the left horn, the direction to where the horn points again changes to now point INWARDS, towards the center-line of the kudu's head. This is the 3rd turn on this particular kudu.
Again with reference to the kudu picture posted by the OP: At about the point where two slightly thicker twigs seems to "disappear" behind the left hornthe direction changes again such that the horn now points OUTWARDS away from the center-line of the kudu's head. This is the 4th turn on this kudu's left horn. So this kudu is a 4-Turn kudu! A good well aged specimen of which the horn-tips point decidedly OUTWARDS.
Just for completeness of my reply, please note that both CAustin's kudu, as well as the one shot by his son, are 3-Turn kudu, in which the final horntips point INWARDS.
Summary: The term "turn of a kudu horns" refers to the change of "pointing" direction from towards the OUTSIDE to the INSIDE, and visa versa, of the center-line of the kudu's head.