There is a special tool you can buy for turning a strap.
You can do it sewing the end then pushing it over a pencil--
or my trick is to cut the strap a bit longer and securely sew a long piece of twine on one end in the fold. Then fold over and stitch just like you were making cording. Pull the thread and it turns inside out Then cut off the extra on the end where you've sewn the thread.

Good suggestions, oilpainter,

If you want to use the old pen trick here's how. Take a safety pin and pin it to one side of the finished end. push the seamed end over the pin and slide the material of the strap down over the pen carefully. Once you get it started, it goes rather quickly and will soon have your strap turned right side out. It's a little hard to get started because of the seamed end but it will begin to turn.

That little tool is called a bodkin. If you use it, you're limited to making tubes large enough for the pincher teeth to fit through. For narrow tubes, I use the safety pin method described. For VERY narrow tubes, I use oilpainter's method, but I use unwaxed dental floss.

All methods require patience.

You can buy a loop turner now and make the finest of straps and loops; I love the thing. Unlike a bodkin, it is a long, thin, needle-like affair about 8 inches long with a tiny clasp on one end and a circle to pull with on the other. Insert it through the tube, fiddle with the clasp until it latches, and voila! you can easily turn the narrowest spaghetti straps. Mine is a Dritz Loop Turner. I've had it for at least 20 years (the package says I paid $3.95 for it) and have turned dozens of straps with it for prom dresses, gowns, kids bows, hair ribbons...

Sherry

The loop turner will turn a much narrower strap than a safety pin will. I use my bodkin for pulling elastic through casings.