Can we bring bottled water onboard ?
The formal answer to this is no. That being said, Royal usually tends to overlook it when people bring bottled water onboard.
Keep in mind that you will need to bring it with you in your carry on luggage. If you put it in your checked luggage, your bag will probably be flagged as it may look like alcohol bottles on the scanner.
Another option is to bring refillable water bottles on board. There are several different water stations around the ship, so its easy to fill them, just remember to use a clean glass to get the water and then pour it into your bottle.