What the others have said is true. However, I'm a lower level rider myself, and the horses I ride generally are not well-trained horses. My own horse can be trained the way I like, but other horses might not be so willing.
Also, since I ride my own horse about 3 hrs/week and have an old back injury, we've modified things a bit. I'll share it with you as food for thought. I think it is important to always remember that horses respond in the way they have been trained, and not all are trained the same way.
When I took riding lessons, the lady who taught them taught turns as "Seat, leg, reins if needed". That is fine, but my seat is not very consistent or sensitive. I may wiggle around to relieve tension in my lower right back, or twist around looking at something and want my horse to ignore me...and how fair would I be to tell my horse, "Turn off of my seat and balance except for all the times I don't want you to do so"?
So I'll still look in the direction of turn, but I don't really expect my horse to turn off of that. I then nudge with the outside leg. Assuming she isn't too excited to listen, she'll make a medium turn off of that. If she is a bit more excited, lazy or I need a tighter turn, I'll add neck reining to that, or an opening rein. My approach is western, and it is my job to keep slack in the reins unless my horse forces me not to. So neck reining would be moving my one hand with the reins over 4-6 inches and letting the outside rein rest against her neck. If I am using two hands, then it is my responsibility to keep slack in both reins, and I'll move my right hand 3-6 inches to the right (and not back...if anything, slightly forward) and invite a right turn (having first asked with my leg). That will generally buy me a sharper right turn.
If she is excited or I want a tight turn, then she'll get steady pressure with my left leg, my right hand will go out about 6 inches and I'll use my pinkie to pull back on the rein. Worst case, I'll take out all the slack and pull harder. I can't remember the last time I did, but it remains an option if needed.
This approach may differ greatly from others. If you were going to ride my horse, I'd tell this to you in advance so you would know what she was expecting. One of the horrible things we humans do to horses is expect them to respond the way WE were trained, and not ask if they had different training - or none at all.
Most of the lesson horses I rode were too dull from beginning students to turn with anything less than hard leg pressure. The poor animals had so many conflicting signals given them that they had largely turned off their receivers - and who could blame them? However, if ridden consistently for 30 minutes, you could see them begin to recognize that maybe this rider could be listened to...
Your instructor is right. The posters above me are right. So am I, although I do it a bit differently. How subtle you want the horse to be depends on how subtle and consistent you are willing to be. VS Littauer was an English riding instructor who taught jumping. He deliberately taught 3 levels of control depending on how experienced the rider was, how frequently the rider rode, and how much subtle control was needed to get the job done. He would have placed someone like me, riding 3-4 hours/week instead of a day, at the lowest and simplest system of control. That isn't meant to insult me, but it recognizes that it isn't fair to ask my horse for refinement that I don't practice often enough to support.
The right way to cue a horse to do something is a balance between the horse's training, the rider's ability, their time together and their goals. A caring rider thinks about those things, and modifies their style according to what they can properly ask of their horse. He should also recognize that a strange horse may have never been taught the cues he uses and be willing to adjust accordingly. IMHO.