In this video demonstration I'll show you how I stop the twist when making vinyl straps. Here is the full video.
I'm using our Thermal activating vinyl here.
I usually cut my straps double width of what I want the finished size to be. For example, if I want a 1" wide strap at the end, I cut a 2" wide x length strap to start with. Interface the entire strap with SF101 or similar interfacing, and do not pull or push when you're sewing. At all.