Mix water and egg whites in the bottom of a mixing bowl. Let stand 3-5 minutes. Add lemon juice or cream of tartar if you choose. Beat on high speed until stiff peaks form. Add powdered sugar one cup at a time and mix well. The resulting icing should also be about stiff peak thick. (Because I like it that way. If you don't like to make it stiff first, then don't add all the powdered sugar.)
NOTE: The lemon juice and the cream of tartar are for YOUR protection. They stabilize the egg whites. If your beat egg whites too much, their protein structure will be compromised and they will sink and get all "weirdy" and then you will have to get rid of them. And probably cry. And then try all over again. BUT if you add lemon juice or cream of tartar, they will beat at stiff peak phase for a while longer, giving you time to realize they are ready even though you got totally distracted by the firetruck that drove into the river across the street. Personally, I love the flavor the lemon juice adds to the royal icing so I almost always add that instead of cream of tartar. Plus...let's be honest. It's WAY cheaper. And you can judge me if you want, but I'm totally the kind of person who cares.