Probably because you need to be 18+ to apply thats why its signing you out. But its also signing me out but I think thats because my DOB is wrong or its on my brothers DOB. Idk try making a new account and do 18+ make sure to actually be 18+ I think you can get your parents to do it as well im not to sure
Try doing it in an incognito browser. Also FYI you need to have spent 21 of the last 90 days in Fortnite creative to apply.
It was doing the same for me. I had to use a different internet browser and it worked.