Does the FAA dictate that bridges are closed for the Thunder Over Louisville airshow event?
The FAA has no authority over roads, bridges or any other public roads. Usually, it's the the event organizer who's responsible for requesting and coordinating road closures; think of triathlons, parades etc. And even without the FAA, there are very good reasons for the local police to close the bridges: rubbernecking and accidents caused by distracted drivers; keeping people safe when fireworks are set off from the bridges and so on.

But, at least in theory the FAA could indirectly require road closures as part of an airshow waiver.

Most airshows require a waiver, i.e. permission from the FAA to 'ignore' certain regulations. The list of regulations that can be waived is in 14 CFR 91.905 and you can immediately see why some of them would be waived for an airshow. These ones in particular seem very relevant:


  91.111   Operating near other aircraft.
  91.117   Aircraft speed.
  91.119   Minimum safe altitudes: General.
  91.303   Aerobatic flights.  


The full instructions for issuing a waiver are long and detailed but there are lots of requirements to ensure minimum distance between aircraft and spectators, and to ensure that the energy vectors of aerobatic maneuvers are not directed towards crowds.

I haven't been to Thunder yet so I don't know to what extent the aircraft actually maneuver vs. flying in stable formation down the river. But, it's at least very plausible that the FAA has issued an airshow waiver requiring a minimum distance from spectators that can only be guaranteed if some of the bridges are closed. And, if the park area around River Road is the main spectator area, then any aerobatic maneuvers would have to direct their energy vectors along the river, i.e. towards the bridges. That could require the airshow organizers to request that the bridges be closed.

Unfortunately I couldn't find out if airshow waivers are public or not (a quick search on faa.gov didn't find any) so I don't know if it's possible to check which FAA regulations have been waived for Thunder, or what conditions they placed on the organizers.