What you want to do for this type of arm wrap is to make each “wrap” as a single piece, rather than trying to wrap the fabric every time and hope it stays.
There are two main methods of doing this: you can either sew the strip of fabric down to a tube of fabric, making sure there is enough stretch in the fabrics themselves to fit over your hand and arm, and then simply slip your arm into it.
The other method is basically the same, but if you have non-stretch fabric, you will need to create some sort of closure, like a separating invisible zipper, on one side so that you can take the item on and off.
I hope that helps! Good luck :]