Just for future reference: Normal practice in the US is to simplify wood size fractions when possible. So 3-6/8" becomes 3-3/4".
Also, most of the time with this type of work you want to use screws - drywall screws for drywall into wood and wood screws for wood into wood (in a pinch drywall screws will work for anything, but they are not ideal except for drywall).
So if you don't already have one, pick up a battery powered drill-driver. 18V-20V is the typical these days. At Home Depot the moderately priced brand is Ryobi - typical is a kit with basic drill/driver and battery and charger currently for ~ $60. Get a pack of drill bits and driver bits (the brand does not need to match the brand of drill - whatever seems like a reasonable price that day). Also pick up a hand saw for cutting wood to size - you do not want to get a circular saw to cut a few occasional small pieces of wood - too dangerous if you don't use it enough to get to know it, and really not necessary.
There are a few ways to deal with this. I 100% agree with "towel bar into wood", but I wouldn't try to use tiny pieces (2" x 2"). Two possible fixes come to mind:
Find the two nearest studs. They are typically 16" apart (on center) though it can vary. Cut a piece of wood, typically 3/4" or 1" thick, long enough and wide enough to reach both studs and to hold the towel bar where you want it. Optionally paint or stain to match the room - easiest is usually just to paint it white. Screw it into the studs. Screw the towel bar into the wood. If the wood covers the existing drywall holes then you don't even need to patch the drywall.
Cut the drywall hole so that it reaches a stud on each side. Cut a piece of wood, typically fairly thick (e.g., a piece of 2x4, which is really 1.5" x 3.5") to fit exactly between the studs. Screw it to the studs (this requires screwing at an angle, so a little trickier than an outside fix). Cut a piece of drywall to fit the open space and screw it into the wood. Patch/paint - how much depends on how nice you want it to look. Install the towel bar, making sure that the screws go into the wood.
You can also cut the existing hole larger, but not all the way to the studs, and use layers to get your 3-3/4" depth. A 2x4 is nominally 1.5" x 3.5", so a piece of 2x4 screwed to the "beam" plus a piece of 1/4" plywood should get you to 3-3/4", though you may need to measure carefully and you might actually need 3/8" plywood. But I would go with either an outside piece of wood (avoid the drywall work!) or inside stud-to-stud (which is the standard way of doing this).