Now that you've passed over the guards and collected the nearby Runes and Bone Charms, it's time to head over the drawbridge. Enter the second part of the map via the loading screen. You'll soon hear a member of the City Watch talking to a guy named Pratchett in front of his house. Pratchett is an important target here, so pay attention to him. He has a key in his pocket leading to his house. You might want to save here, especially if you're going low chaos. Let them finish talking, and stalk Pratchett. Choke him out, hide his body, take out the member of the City Watch, and steal the key from Pratchett. Now, use the key to get inside of his house and search the safe. The combination should be 473. Nab the Rune from inside of it.
With that out of the way, leave the building and head towards your waypoint. Along the way, stop next to Pratchett's house to find another Rune. You will be able to hear a screaming nutjob next to Pratchett's house. Use Blink to get into the guy's house, take him out, and steal the Rune from his Outsider Shrine in the corner.
Now finish your trek to the bridge. You'll find an Arc Pylon here; this is nothing to mess with, as it will fry you in seconds flat should you approach. Take the side route next to Pratchett's house to get to the bridgehead, since the area below is occupied by enemies and the Arc Pylon. When you reach the entrance to the bridge, you will find some thugs talking about messing up the City Watch. Stand there and let them do it. With the thugs fighting the City Watch, quickly Blink or stop time to rush past them and they won't even see you coming. Now you are officially on the bridge.
Continue under and around the bridge, avoiding the numerous Arc Pylons there or powering them down by following the wires leading from them back to their power sources. If you are trying to play with clean hands and ghost, it's definitely better to skirt the edges of the bridge using Blink, as going for the power sources will inevitably lead you into some form of contact with enemies. Move along the outskirts of the bridge and Blink up to the control booth to raise the bridge. Use the chains dangling outside of the control room to circumvent the guards as they come looking for you, and use them to get to the stairwell. Keep following the chains to find a corpse, a journal and a few items at the bridge's highest point. From here, you can pretty easily Blink to wherever you want on the bridge; remember that if you jump, it can extend a Blink's distance by your jump's length. Save before trying this, because it's really dangerous. As you reach the other side, you will meet up with Alec the Scavenger.
He will be trapped, and beg you to free him in exchange for half of his share of River Krust pearls. This optional objective won't have too big an effect on the rest of the game, so it's up to you whether or not you want to save him. If you are trying for a ghost or shadow run, definitely don't do this mission. You'll be ambushed as a part of the mission, so it's essential you don't do this. Otherwise, nab the key to free Alec from a table next to the cell. Choke out the guards and use the key to free Alec. He will lead you down a side path blocked by devious River Krust enemies. These acid spitting critters have high defense, are nigh invulnerable when their shells are closed, and can deal a lot of damage from a distance with their projectiles. They may not seem it, but they're by far the most dangerous enemies in the game, since you can't avoid them as easily as a tallboy.Take out the River Krusts when they open their shells, and then loot them. Alec will ambush you with his crew, leading to a fight to the death. Unless, that is, you use Dark Vision to scan for enemies before you go through the gate. You can Blink into the room and take both of them down, raid the stash, and meet Alec's taunt with a sleep dart or a pistol bullet to the face.