The river mentioned in the name is the one south from the Brass Fortress, with the two bridges on either side. If you are standing on a bridge, you’ll need to make a little effort, to go down to the lower level. The bridge is a lot higher than this ground level, where the river is, and you might consider not jumping off it. When you find yourself down there, you won’t have any problem finding the skill point piece. It is close to what I presume is the main source of the river’s strength, the town’s sewer exit. It is not the best looking location, but this is where the drainage pipe Skyshard is.
The first shard on the list is found near the main bridge that takes us into the Brass Fortress, north from the main Clockwork Crossroads Wayshrine.