Foil boards can carve and they can pivot.
Pivot turns are the narrow 360s for instance, some do these without leaning back, simply a pivot turn.
Then you can also go into a carve with speed, leaning over, and pushing hard - just like you want to.
Best results are with a wide and at least rear foot back stance, and you need speed.
The popular really big wings many use (me included when waves) are no good for this, they simply go too slow, so you lose some of the carving feel.
BUT, they can turn so narrow, that it is sometimes even more fun - and you DO really carve when you ride 180s or more, downwind, with these wings also.
A bit smaller and faster wing, is in my experience better for carving hard and getting the "laydown" pushing feel.
Ride with speed, lean hard into the turn, and carve hard - as said, you must not use too short a mast, and the more speed the better.
Also, flying the kite really low when initiating the carve, makes wonders.
Besides the way you do it, foils also play a role, meaning if you got a supersmall stab, you lose some of the driving feel.
Flat wings and flat stabs also drive less joyful IMO.
Exactly the same as having a surfboard with too small fins, you lose the driving carving lovely feel.
They are loose and twitchy yes, but no drive.
Too big a fin, and it feels locked and boring, so one has to find his/her personal liking, both for surfboards and hydrofoils.
You have no spray in a cutback on a hydrofoil, so no matter how hard you push, it still feels fluid and great but extremely different from a surfboard (or TT) spray
Carving hard frontside though, is the same great feel in my experience