Let us say we have a large, late medieval era trade city that is also a port, sited on an estuary. Due to the city's narrow, winding streets, caravans are not allowed within the walls: cargo must be portaged through the city on foot or the backs of small pack animals such as donkeys. Furthermore, not only are there roads and seagoing ships going in and out, there are tradeships that ply further upriver, carrying loads far in excess of what even a caravan can.
However, the volume of road trade is overwhelming the portage system and leading to caravans camping for days, even weeks, just to get from one side of town to the other. Worse yet, there are no bridges within town, just ferryboats, as the estuary itself cannot be bridged without the invention of caissons.
This near-gridlock has led city leaders to consider the seemingly unthinkable -- bridging the river upstream as part of a bypass road project, as it is much shallower and somewhat narrower there. However, there is a complication: this bridge is being built without the benefit of high banks to elevate it, yet must clear the masts of tradeships 20m high while providing at least 30m of clear channel width and preferably as much as is feasible with the materials available.
How could our medieval bridgebuilders build a movable bridge that can handle this?