In your experience, does a crosswind have a similar effect as a headwind in making the disc fly as though it's experiencing higher airspeed? I would think that a disc would act no different due to aerodynamic HST in a 15 mph crosswind vs calm conditions.
So consider a RHBH throw.
Left to right crosswind will push a hyzer shot down because the top of the flight plate is facing the wind. A flat throw with a true stable-overstable disc should go pretty straight still...think Teebird/Thunderbird. But throwing 100% flat and with wind variance along the flight...at some point the underside of the disc is going to see some wind. This will push the disc to the right. So a more overstable disc will fight this L/R crosswind and hold straighter, eventually getting a fade and as soon as it begins to fade the top of the flight plate gets pushed down by the wind so it'll end pretty straight. If you throw an understable disc in a L/R wind, once it gets just past flat the disc will get carried right and it will never have the added stability to fight the wind at all.
So basically in a L/R crosswind you want stable-overstable, ideally for distance just enough HSS wiggle that the crosswind is pushing the underside of the disc the whole way while the disc is overstable enough that it is trying to track left, keeping it gliding. If you go true OS like a Firebird, it will hold dead straight and then as it fades it gets pushed straight down so the fade time is really decreased, almost like a Teebird shape. If I had to make up some numbers that are
pretty much guesses based on my experience, the flight path for many of my discs with say a 10MPH L/R wind are like a 15-20MPH headwind. However the distance is very different...crosswinds don't kill distance and sometimes you get great carry, whereas headwinds tend to eat up distance unless you are throwing a disc that is too fast/stable for you and the wind kicks it into its intended flight characteristics...not common but it happens.
In a right to left crosswind, a stable-overstable disc will show the underside of its flight plate as soon as it starts to fade, and it'll get pushed way left and pretty early usually. Typically a moderately understable disc is best thrown on a slight hyzer flip. As it flips to flat it does its normal thing, then as it tries to barely get past flat the wind holds the flight plate down level and it just tracks straight. Then since it's a mellow disc it won't have a ton of fade, but that will still get accentuated by the wind carrying the underside of the disc left at the end of the flight. If you go super understable it can get pushed down as it flips over too quick.
So really it's not that it has more turn in a crosswind, it's just that as soon as the turn exposes the bottom of the disc you then have to ask how much stability the disc has in it to fight the wind that is now trying to control it.
Edit: This is all based on my experience throwing windy rounds and practice sessions.