A lot of girls have mentioned that they don't want to mess with their leg or facial hair because they're worried that the regrowth may be thicker. Well, I'm happy to say that's a myth! The hair that you see on the surface is already dead. Shaving or cutting it close to the skin does not in any way affect the live hair growing out.
The regrowth looks thicker and feels scratchier simply because it's been cut near the base. Therefore, it looks darker, compared to the tapered fine tips that you just removed. :) The longer shaved hair grows, the softer it becomes, so you don't have to worry about the effect being permanent.
What a hair follicle looks like - thick near the base, tapers at the tip
Waxing, threading, or plucking involves pulling the hair directly from the roots. When it grows back, it will come out with tapered tips, the way new hair does. It won't look thick at all unless you or your aesthetician did it wrong by ripping off the hair mid-shaft instead of the roots.
Lastly, according to Newsweek, regular waxing can even lessen hair growth over time. The process involves traumatizing the roots, which may then eventually close up. Laser is a quicker and more permanent solution - a costlier one, of course, but at least you don't have to do it regularly.
So yes, don't worry about the hair regrowth! :) I've already shared my views about body hair so you can take a look at that post. Hope this helps!