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It seems that the screws attaching the towel rack to the wall were not driven into the framing, only into the drywall. After years of use they were bound to come loose. The solution is to provide a solid backing for the screws.
Here's how:
Cut a 6-inch-square piece of drywall from the damaged part of the wall. Make sure the failed screw holes are in the center of this piece. We recommend that you make these cuts with a utility knife. Although you'll have to make several passes on each side of the square, you will get a clean cut. This is important, because you will save this piece and replace it once the wood backing is installed. This ensures an accurate fit to the patch.
If the piece you remove is too badly damaged you may have to cut a small piece of drywall to fit. The problem here is that drywall comes in 4-by-8-foot pieces. Don't buy an entire piece.
Drywall gets damaged regularly at home centers and lumberyards. Check to see if they have a damaged piece they will sell you at a reduced price. Or better yet, ask if they will give you a small piece salvaged from a severely damaged sheet. We've even been known to Dumpster dive at construction sites.
Once you have a hole in the wall, cut a piece of wood so that it is a little wider but not as long as your 6-inch-square opening. An 8-inch by 5-inch piece of scrap ½-inch plywood will do.
Drill a small hole in the center of the wood and thread a string through it.
Knot the string or tie a nail or metal washer to it so that the string can't be pulled out. Make sure you can work the wood through the hole in the wall and that it can be pulled up against the back side of the wall by the string.
Once this backing is prepared, apply construction adhesive around the edges of the plywood and, using the string, pull it securely against the back of the wall. If there's enough room to get a finger or two behind the plywood, you won't need the string.