I actually had a chat to the swim school about the face under the water thing for DD2. She had a teacher once that would have them on their hip and go under with them and DD2 was petrified of it. We ended up speaking to the teacher and getting her to take the soft approach with her. The teacher did try but if she didn't want to then she didn't force it. She would start with "can you put your chin on the water?" and if she did yay good job. Or your mouth in the water and blow bubbles etc. That worked much better for DD2. She was still a bit scared but she stopped crying about it.
I tried a couple of teachers during the school holidays when DD1 first started to see if those teachers worked better with her. Sometimes it can be a personality thing too.
With DD2 when she was scared of jumping in (She used to scream she didn't want to die) it was hard. Her teacher had a talk to her and came to a deal with her. She used to hold her hands and just get her to do one at first and promise her she wouldn't let her face go under which helped for a while.
Once her actual swimming got better the swim school suggested a change of teachers just to get her away from the teacher that kind of brought up her memories of when she first started. That worked well along with a chat with one of the other swimming teachers that is usually in the office. DD2 actually told me one morning she didn't want to do lessons anymore and I told her to go tell them so she went and lined up! The lady took her into the office and had a chat with her. Apart from explaining to her the importance of learning to swim she told her she wouldn't be happy with letting her swim on her own when we come to the pool. She's the one who suggested a change in teachers and we actually tried one that day and it worked well.
Now that DD2 is swimming better she needs a firmer teacher but it's different for her now that she swims the 25m. Sometimes DD2 will stop and have a bit of a panic and her teacher will say what are you doing, keep going. Her teacher said she can see even now that she is a bit nervous swimming even though she can do it.