I have always been curious about the practice of making a 3 gallon wort and then topping it off with water to make 5 gallons to ferment. I might be doing it wrong, but I add my cold, filtered water into my primary and then add the chilled wort into that. I have always been curious why this is done this way, it seems like regular filtered tap water, might effect the taste or worse yet add unwanted bacteria into the mixture. Am I doing this wrong, or why is this not a concern?
Bacteria typically isn't an issue as there shouldn't be anything terribly nasty in your tap water, and once the yeast take off they tend to take over. That said, I know a lot of people use bottled spring water instead of their tap water, due to taste issues.
I think one of the big reasons for brewing smaller amounts and topping off with water is due to equipment. People are more likely to buy kits if they can brew them on their stove with a stock pot they already have rather than having to invest in a full 10G pot which they may not be able to bring to a boil on their stove, or which they can't cool in their sink.
I buy two gallons of distilled water at the store and freeze it. When it's time cut it open and put the ice in the primary and poor the wort over it. Thats how I chill the wort down before pitching the yeast.
This one might be of interest Ice the wort for fast cooling
I wouldn't use tap water, personally, but I know people who do and they don't seem to have an issue.
Although bacteria is a concern - the bigger issue is chlorine. You can solve that problem by just leaving the water uncovered overnight. The chlorine will dissipate naturally and you're good to go.
However, if it's only 2 gallons that you're adding, why not just boil that amount the night before so it has time to cool and use that? Marginally more work but you won't have to worry about bacteria OR chlorine that way.