How close can I fly to a bridge?
If operating under VFR rules, which your pilot probably is if he's attempting to stop you actually hitting the bridge in question and allow you good enough visibility to take a photograph in the first place, the rules vary slightly by location. 

The FAA regulations will be as follows (Note: all emphasis mine - indicating what I believe is the interpretation of your bridge)
:


  An aircraft must maintain an altitude of 500 feet above the surface [and], the aircraft may not be operated closer than 500 feet to any person, vessel, vehicle, or structure.


and


  An aircraft must maintain an altitude of 1,000 feet above the highest obstacle within a horizontal radius of 2,000 feet of the aircraft over any congested area of a city, town, or settlement


Also note that helicopter rules differ slightly.


  Helicopters may be operated at less than the minimums prescribed in [the previous quoted sections] if the operation is conducted without hazard to persons or property on the surface.


source: 14 CFR 91.119 - Minimum safe altitudes: General.

And the UK CAA rules are similar


  An aircraft must not fly closer than 500 ft to any person, vessel, vehicle, building or structure.


and


  [...] it must not fly less than 1000 ft above the highest fixed object within 600m of the aircraft.




So, in summary your pilot can get you within 500ft if it is an uncongested area, 1000ft if its a congested area - closer if he's got a helicopter and can do it without causing a hazard.