You are probably carrying too much speed into the turn.
First - Are you cutting to the outside after crossing the wake? If you are, the spray from the ski will be heading towards the boat. This is a common error with new skiers (I know you are just learning a deep start, but I don't know your skiing ability - so forgive me if that sounds too basic). Try to cut across the wake and do not cut after crossing the wake. Instead, get the ski on the inside edge (right side on a right hand turn)as quickly after crossing the wake as possible. Don't try and turn right away, just get it on the inside edge. Then conscentrate on coasting and slowing down - when the ski slows down to a point where it is almost sinking - then finish your turn. You may not get as wide at first, but that will come when you learn to cut harder. Too many people will ski like this: they gingerly cross the wake, then cut hard, then try to turn and gingerly cross the wake. This is how you should do it: cut across the wake, coast after crossing and slow down, turn the ski, pull and cut hard across the wake, coast, turn, etc. If done properly, the spray from the ski will ALWAYS be away from the boat.
If you are doing this and still feel like you can't turn, be sure you are bending at the knees. This is what I mean: you should keep the same good body position (hips forward, handle into hips, shoulders back) but bend the knees and push them forward. You will end up leaning a little further back to keep your balance (you are not leaning forward on the ski, but just moving the knees forward). This will slow the ski even more and allow you to turn it better.
Everyone usually struggles with their off-side turn. It really takes good form to do it correctly.
Eventually, you can start to make small changes on the fin and bindings, but I wouldn't do this until you are sure your form is 100% correct. Most skiers won't change any fin settings until they are skiing in the slalom course and have really shortened the rope (28+ off). Until then, keep working on the form.