I have tried many methods to turn narrow tubes of fabric, including borrowing my mother's FasTube turner set. The one that has worked best for me with the least amount of fabric distortion/bruising is to use a ribbon basted to the short end of the tube. I can't remember where I heard about this technique but it always works well for me:
Step 1: Cut a length of ribbon that is a few inches longer than the tube you wish to turn.
Step 2: Fold the fabric in half so that the long edges meet. Sandwich the length of ribbon in the middle.
Step 3: Baste one short end of the fabric layers to secure the end of the ribbon.
Step 4: Stitch the long side of the fabric layers, being careful not to catch the ribbon in the stitching.
Step 5: Pull the ribbon through the tube, thus turning the tube right side out.
Step 6: Remove the basting threads that secured the ribbon. The ribbon will easily pull out.
The job is made easier if a wider ribbon is used for wider tubes.