As John pointed out an airboat hull by design is meant to slide, an as such, has no horizontal resistance to such an occurrence.
This is exactly what you lacking that you need to add. John's method is one such means of making that addition.
When you say you need to make "tight turns" it leads me to think you will ultimately want more 'horizontal resistance' that what the described means will afford you.
I have had better results by adding a pair of keels (for lack of a better word) or external stringers (sounds stupid, but is descriptive) to the bottom than adding a single one up the center line. Basically what I have done, is take a length of wood and rip it diagonally to end up with two stringers to be placed fore and aft on the bottom of the hull.
View attachment 97882
Place these stringers such that they have about 1/3 of the bottom width between them. Note that the flat side goes toward the center.
By positioning them in this fashion the inside of the turn will pull (or grab) on the flat side of the stringer, but the outside of the opposite stringer slides over the water and tilts the hull a little bit but does NOT grab. You want the grab and the tilt to favor the inside of the turn for obvious reasons. High siding a boat is not much more fun that high siding a motorcycle.
As far as the placement is concerned, because they are not on the center line and because the planing surface is all the way aft, the best results will be attained by having them start right at the transom going forward.
I would attach them as John describes with the exception being that a couple full stringer widths of fiberglass mat will allow you laminate the stringer directly to the bottom before glassing over to seal it.