Do airplanes with fixed gear usually turn upside down during water landing? During discussion about emergency landing on water in this YouTube video (4:00) the flight instructor said:


  "...the problem with these airplanes with a fixed gear, you're usually
  going to be upside down".


Is it really like this? If so, is there any water landing technique which could minimize the risk?
Depends on the airplane.  I recall reading about a guy flying across from the Aleutians to Japan, in a Piper Super Cub of all things, and he had to ditch it for whatever reason, which he was able to do next to a freighter.

To his surprise, it didn't flip over but kind of submarined, stopped, then popped to the surface nose down but upright. It's possible being a taildragger helped.  I think tricycle gear airplanes would be more prone to flipping right over because the pitching moment from water contact is a little stronger with 3 gears hanging down vs two.

I expect though, that if you surveyed ditchings you would find some do and some don't.  Pitching moment from water contact, elevator effectiveness in resisting the pitching,  amount of up elevator at water contact, speed at water contact, maybe the surface contours of the fuselage could promote it or inhibit it, all kinds of variables.  

As xxavier says, it would be prudent to assume you are going to flip over, and if you don't, bonus.