Learning to swim is hard because you have to do several things at once or you start to drown. So, contrary to dogma, I got her some flippers and a floating vest.

I don't know what the swimming programs in your area are like, but where I live (under the jurisdiction of the Canadian Lifesaving Society), the kids start out with skills in the shallow end (floats, front and back, first assisted and then unassisted, kicking on noodles, dunking their heads, etc) and as soon as Preschool C, the third level, one of the items on the checklist is having them swim around in a lifejacket in the deep end. I don't rigidly follow the checklists because I find it limiting, but most instructors do, meaning that in most of Canada at least, they should be taking 4-year-olds in the deep end with aids.