My dad opened up the top panel of his scooter and showed me what he thought was the problem and as soon as I saw it I was like "yep, 100% that's the problem." The cables coming out of the turn signal switch were installed in a way that pinched the wire between the plug and the bottom of the frame of the brake lever.
This is the turn signal switch
https://ibb.co/GJ8NqP6This is my drawing as a 4 year old
https://ibb.co/cwxbD4MFor now we just wrapped some electrical tape around and it and installed it properly to where the cables run just BARELY underneath the brake lever frame without a problem. We tested the scooter and it runs fine now, no lights turning off after using the turn signal, no blown fuse, WOOHOO! Time for me and my dad ride tomorrow.
Thanks for the help!