Buy a bow (if you don't have one already) and some wooden arrows from the Undead vendor, then go back to the bridge. Get your timing right and run between fire breaths, you'll find a staircase midway through to the right. This area will allow to continue to the next area from under the bridge, and unlock a shortcut that takes you to the last bonfire you rested at. When you advance from under the bridge, you'll find a spot on the right side where you can target the Dragon's tail using the bow in first person view. Aim and time your arrows right and you should "cut" the Dragon's tail after a bunch of shot. This will earn you the Drake's Sword (200 attack), however I don't think you can use it until you bump you STR to 14 (or 16?). Anyhow, it's the best weapon you can find at that point, and it will annihilate anything for at least 6-10 hours of play.
If you find any of this confusing, just youtube "Dark Souls Drake Sword."