There are an immense number of variables. Personally, I would not fret over exactly what my item level is and just enjoy playing the game, letting the equipment upgrades ensue. That said, you could instantly reach the desired item level largely by just purchasing Item Level 220 crafted gear from the market board (if you have the gil, of course). I bought a few such pieces solely for the purpose of increasing the average item level of a newer friend that plays less so that she could instantly meet the requirements of the next dungeon to progress the main quest (effectively skipping a lot of other dungeon runs to collect intermediate equipment and tomestones to buy better equipment).

And also keep in mind that the relic weapon quest is NOT a quick or efficient way to upgrade your equipment. Do it if you enjoy it, of course, but never feel obligated to do ANY content you don't enjoy.