Hats off to you, OP, for listening to your instincts and your dd 
dd2 had her first formal swimming lesson a few weeks ago. She could manage about 10 metres out of her depth, when feeling safe in a pool on holiday, but is really just beginning to learn proper strokes - she's 6.
The wretched swimming teacher barely said hello, had her out of her depth pushing her to do lengths, compared her to another child (unfavourably), and told her she was a big girl and shouldn't be crying - when she inevitably started to panic and give up. It was awful. As she's not my first, and I'm old enough to not care what all the watching parents thought, I tried explaining to him that he was pushing her too fast and that she wasn't that confident/able yet. Then when he got defensive and kept up the 'I think she is fine' and dd carried on quietly sobbing, I told him this wasn't going to work and pulled her out of the lesson.
Two weeks of 'upset tummy' at the thought of swimming lessons, then we finally started somewhere else with a lovely man who let dd go at her own pace, where she can touch the floor, where he talks to her and actually makes a relationship with his pupils. Within two lessons, she was doing a length of back crawl, front crawl, and jumping in - totally different experience.
I've come across the odd 'bully' type swimming teacher - and lessons that feel like 'bullying' no matter how well meant. I'm not an advocate of teaching through fear - overcoming fear takes support and a sense of being a little bit in control, not force. I wish I'd known about MN and been able to get this kind of support as a parent when dd1 was teeny and hated swimming lessons - had to get there by trial and error 