How powered are you? Generally for the average foiler, the more power, the wider the radius you need to bleed the speed. The more proficient you get, the more you can oversheet and really lean into it though. If you initiate a harder carve from the beginning, you can feel the power shut off very quickly assuming you aren't wildly overpowered. It takes a bit of commitment though and for a lot of intermediates, it's uncomfortable to be that lit on a reach and rail it over pretty aggressively.
fwiw, I'm usually doing step jibes but I see no reason that you couldn't do strap to strap if that's where you are more comfortable.
In lighter air, you absolutely have to carve tighter as you have just a limited amount of energy to get you to the other side. However, don't carve too high on the other side despite your wanting to stay upwind. The primary goal is to stay on the foil.
I was chasing dinghies upwind yesterday in a really gusty 8-18 on a small board, 5.4 and on a small lake. It took a lot of hard carves to stay with them. In the lulls, though, I was toast.