Cooling a 2,000V laminated steel transformer?
Not distilled water. You may have heard that Sandia insulate transmission lines with the stuff, but that's for less than 1 second, with ultra-pure, cooled water, so forget it.

Oil, that's mineral or vegatable, will cool the transformer and increase its thermal mass to some extent. Whether it will make enough of a difference to be worth the mess and bother is another matter.

The problem with trying to liquid cool a transformer is that if it has not been built for that, it will not have gaps between the windings so you can get the oil into direct contact with what's getting hot. Indeed, in a transformer built for conduction cooling, there will not only be no channels to get fluid through, but no voids between layers.

An air-blast may be almost as effective, and a lot less messy, than oil.