The National Bridge Inventory reports that this bridge carries "US 23S" and classifies the route as a "U.S. numbered highway spur". Presumably the "S" stands for "Spur" and not "South". Bid documents by the Kentucky Transportation Cabinet also refer to the bridge as carrying US 23S. For example:
http://www.kytc.state.ky.us/Contract/Lettings/2006/1027/BidD...
Moreover, the route number of "23S" appears in the bridge's inventory number: "010 023S B00040".
Great website, very useful information. I have a correction for you, however. This bridge is NOT on US-23, but is actually on US-60. Specifically on 13th Street. You can check a map closely, and there is a photo at http://www.kentuckyroads.com/images/ohio_river/.
I checked the American Association of State Highway and Transportation Officials (AASHTO) records and found that both bridges are indeed designated as U-S 23 Spur (since 1986), dead-ending at U.S. 52 on the north end of the bridge.
The link is: http://www.us-highways.com/spurus.htm
But, the U.S. 23-S designation is not used and may be truncated permanently, because of, as you speculated, its confusion with U.S. 23 South. Since there are no signs for U.S. 23-S in Ashland (I didn't spot any when I took the pictures), there isn't any confusion.
But AASHTO indicates, neither bridge carries U.S. 60.