If your scales get thick, you can lose hair for a while. But it usually grows back once your psoriasis gets under control and your skin heals.
If you’re using salicylic acid as a treatment, it may cause hair breakage and loss. But it’ll stop once you stop treatment. The same thing can happen if you’re taking a retinoid.
As a general rule, be gentle when you get rid of scales. And in the meantime, feel confident as you are. Or if you want some camouflage, use it as an opportunity to rock some new headwear, be it a hat, scarf, or wig.