mounting on exterior walls

Just to clarify, by exterior wall I mean the inside of a wall that goes to the outside, not one between the bathroom and living space or something like that

On my last trailer I mounted towel bars to an interior wall just using very short sheet metal screws. It worked fine. My question is about doing the same on an exterior wall.

I am assuming your talking about interior walls.  there are a few different ways to do it.  to me the best way is to get a stud finder and use painters tape to mark where your studs are then get a towel bar that matches the spacing.  there are kinds where you cut the bar to size.  this way you can use a 1" wood screw, predrill and it will be solid.  the other way is to use some sort of wall anchors and drill just deep enough to ge tthe anchor in and then screw to that.  just look at your wall thickness then choke up on the drill bit or put masking tape on it so you know to only drill untill the tape hits the wall.  

Depends on your level of skill and ability to measure accurately. Assume you mean the risk of poking thru with a screw. 
Thats easily avoided with a little measuring and some 3rd grade math. 
just becuase that Temu towel holder comes with 2” screws doesn’t mean those are the ones you can use. 

PS, I need to learn the trick for getting suction cups to stick to textured surfaces….lol

Use suction cup mounts.