Before going to fight the fire dragon on the bridge, make sure you buy and equip your bow/arrows. You can buy bow/arrows at the first undead merchant.

After you beat the Taurus demon, keep going down until you see the dragon swoop down and blow fire, which kills all the archers on the bridge.

After they die, stay close to the right side of the wall and run to the opening on the right side. Go down to the lower level so the dragon cannot kill you.

Now you should be under the bridge. Stay on your right side and keep going. You should see the dragon's tail wagging.

Iused standard arrows so it took me about 20 shots to do this. There are two monsters with shields under the bridge which you can kill or just hide so they don't see you.

Keep shooting the dragon's tail and it will eventually break, giving you the Drake Sword which is the strongest until you move further in the game.