GFCI installation in bathroom switch box doesn't work as expected
You just tried to hook up an outlet to an old-style switch loop with no neutral wire.  Right now, your hair dryer is wired in series with the light, and the switch is dangling in the electrical breeze.

(In other words: You'll need to find another place to put the GFCI function, or run a new /3 NM cable to the switch box so you can have a neutral there.)