What you do is, you dry up the river by diverting it (let someone else worry about that part). Then you pile up a bunch of compacted sand and rocks, then you build the bridge over the compacted sand and rocks, then you un-divert the river (the same guy who diverted it can do this) and allow the river to wash away the compacted sand and rocks which you helped by boring holes through the sides of the pile. Easy-peasy...
In this instance it's not a river.
It's an estuary or strait that joins the Black Sea with the Sea of Marmara.
Aside from the very strong and rapid currents, the depths, 120 ft min to more than 400 feet make it extremely difficult if not impossible to divert the marine passage.
The use of caissons such as pioneered in the building of the Brooklyn across the East River of NYC would not have worked as compression sickness occurred working at depths less than 80 feet.
Those conditions, likely unknown to Da Vinci, also make the use of stone construction supporting scaffolding hugely problematic. While at various times chains were used across the strait to block naval intrusions I doubt the powerful currents and tide would allow for anchored barges to support construction scaffolding, not to mention the difficulties of transporting and lifting the stones, but with Da Vinci who knows.
Just as with construction of the first cathedrals there would have been many failures and deaths along the way.
Check the current bridges...long span suspension bridges, and one of which IIRC is a swing bridge.