With pictures other MH owners would tell more, but....
Other than a log cabin :), any wall has an interior panel with studs behind, and then an exterior panel. "Flimsy wall" that you see, must be an interior panel, often a thin 1/8 plywood. Find the studs and mount on them. In the worst scenario - if there are no studs - glue on a reinforcing plate with a panel adhesive and then drive the anchors in. If you can get behind that wall, than you can reinforce it from behind. But I am sure there are studs at least in the corners.