That's a great story! Your kid is really thinking hard.

The reason that water puts out many (not all) fires is that it cools the fuel below the temperature needed to maintain the combustion reaction, and most common fuels do not have an exothermic chemical reaction with the water. While there is a small amount of molecular oxygen dissolved in most water, the bulk of the oxygen is already "burned", chemically combined with hydrogen. The fuel would have to be able to break the chemical bond between the hydrogen and the oxygen in order to re-use the chemically-bound oxygen in water for combustion.

Not all chemical reactions that generate heat and light (the sorts of things we like to think of as "fires") require oxygen. Magnesium is a great example of a fuel source that can burn under a multitude of circumstances (under water, in a pure nitrogen atmosphere, etc.), which would extinguish most other fires.