I didn't post a full answer before, because I didn't have the time to expand on one. Aaronut spared me time by writing 2/3 of what I would have (citric acid, citrate) and presenting it very well. I won't repeat that part. But there is another point which I find important:
When you say "tangy", do you really mean "sour"? Because for me, "tangy" means a combination of "sour" + "astringent", with sometimes a hint of "bitter" thrown in.
This means that even if you drink pure lemon juice, your drink will still not appear tangy enough to you. You need to add astringency.
The way commercial drinks add it is by carbonation. There are systems you can use to carbonate your own drinks at home, but you'll have to make the investment first.
You could try making your lemonade with store-bought carbonated water, but given the price difference between carbonated water and store-bought lemonade on the one hand and tap water and carbonated water on the other hand, it will probably not let you save much money. It will, however, help you drink better quality lemonade (as in, made from real fresh fruit instead of synthetic flavors).
Another way you can achieve it is fermentation. Don't let it go on too long, you don't want an alcoholic drink. The trouble is, with wild cultures you never know when a batch will turn out good and when it will have off-flavors.
You can also add ingredients which are by themselves astringent. Chokeberries resp. their juice would be a perfect choice for a lemonade, if you don't mind the red color. Quinces shouldn't add much color. There are no other easy astringent ingredients I can think of right now, except for a tea of oak bark, but the taste will need lots of getting used to if used in a lemonade.
If you find out that it's the astringency you've been missing, you probably need no additional citric acid.