i work for a contracting company in which we have a plumbing division which also deals with water purification, filtration etc.. we have done a lot of research on the water being available in normal household and other places which is most commonly used for our basic needs. but talking about this (tap) water being pure.. well i would comment it is indeed pure, after all it is purified by a number of chemicals and toxicants.
but for sure this water is not healthy enough to drink or to use for any other cooking purposes.
anyhow, brushing with this water is not an issue.. because in that case the water is does not reaches your stomach.

but yeah, if your looking for purifying your tap water, it is possible. i know a product name "AQUA FREE" which is also passed and recommended by the Kahramaa authorities to purify the water. this specific filter is as small as a your palm but it has got seven membrane filters installed in it which purifies the water one after another, and unless the water doesn't gets completely purified it does not allows it to pass through it. the water filtered from this filter is as pure to even feed an infant.

thanking you.
hope the above information was of good use for you.

for more details contact me.