The winter turnout actual has a fleece lining. I started using the nylon sheet, because someone suggested that the fleece may have actually caused the hair loss since it may rub the hair rather than sliding over it the way a smooth material does. But, turns out, I have the problem even with the sheet underneath.
So, perhaps I should look into a different turnout blanket, such as the Rambo Alice suggested. I imagine I could find someone to buy my current turnout, it's still in great shape.
Thanks for everyone's help!