But then it would fall apart because of the glue, and it would be all messed up and wouldn’t fit right.
I’m not saying you need to fix it if your current axle is now working… but jhb is exactly right.
Superglue is incredibly thin. Chances are good that the axle broke in such a way that it’d be easy to piece it together like a 4-part puzzle. 
Superglue the 4 pieces together. The tricky part is not accidentally gluing them to your fingertips! Hahaha!
If they fit together like a puzzle, there won’t be any visible cracks. [edit: meant to also say here, and you can sand off any excess that is squeezed out]
Even with thicker white/wood glue, woodworkers often join pieces of wood together, and the join itself (as long as it’s not end-grain) is typically stronger than the wood, while also being seamless.
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As to gluing the pieces back together, do them two at a time, then after each has set, put the two together rather than trying to align 4 pieces at once.
Alright so I’m not sure why but now my Turner’s so smooth! It’s not vibing anymore! The strings all last around a day and a half of play, which is a lot better than before. I am going to need more strings soon though, because the cotton strings I made are horrible.
Cotton doesn’t last on a fixed wooden axle.
What would be best then? Or did you just mean that in general?
There’s nothing better. Just stating the facts to set expectations. 
Its a crazy world we live in. When driving though my neighborhood you will see open garages with nothing in them except an electric law mower. It seem like hardly anyone works on their own stuff any more. My garage is packed with every tool I can get my hands on(man stuff). Tinker away kids, learn to use those tools. Hit the goodwill or garage sales tools last forever.
Best, there is no best…
I prefer slick 8 (50/50 cotton/poly) for my for my fixed axle loopers. I like the way cotton plays better, but prefer the slick 8 because it last a lot longer.
How do burns like that happen? also does it have more friction compared to the ones that is not?