A towel bar that moves when it is used can damage paint, loosen its anchor, and eventually fall. The visible set screw or mounting plate is often only part of the problem, so check how the bracket is attached before tightening aggressively.
Identify the movement
Hold the bar and each wall bracket separately. Determine whether the bar is sliding in the bracket, the set screw is loose, or the mounting plate is moving against the wall. Keep heavy loads off the bar while testing.
Remove the trim carefully
If the design has a cover, release the small set screw with the correct hex key or screwdriver. Support the trim so it does not scrape the wall. Photograph the arrangement if several washers or spacers are present.
Correct the anchor
Tighten a sound screw only until the plate is firm. If the wall anchor spins, the hole is enlarged, or the wall is soft, use a suitable replacement anchor or move the plate to solid framing. Check for hidden wiring or plumbing before drilling.
Reassemble and test
Refit the cover and bar, then apply gentle downward pressure without jerking. A towel bar is not a grab bar; if someone needs body-weight support, install a properly rated safety bar with structural backing.