As part of Global Environment Day celebrations there is an organised bicycle ride this Sunday June 2nd starting from Taksim on European side and going to Harem and Bostanci on Asian side. The highlight is that the Bosphorus bridge is closed down to cars for a period to allow the bicycle traffic across :) This and the Eurasia Marathon are probably the only times you'll get to cross the bridge without a car.
The full route is 35km Taksim to Bostanci but anyone can drop off whenever they want (probably best at Harem/Kadikoy and jump on a ferry back to Europe) as there's no pre-registration. Just turn up and pedal. The ride starts from Taksim Gezi Park at 10:30 on Sunday.
Route map: Protected content
Turkish FB event listing: Protected content FB event trans to English from bike rental co if you want to hire a bike Protected content