Istanbul is undoubtedly far safer than most European or North American cities, and cases of mugging and assault against tourists are rare.
For the average visitor, pickpocketing is the main cause for concern: be particularly careful around Sirkeci station, the Eminönü waterfront, the Galata Bridge, and around Taksim (especially at night). Also avoid being on or around the Byzantine land walls at dusk/night. Be very careful, too, on public transport, particularly when it is crowded. If you feel anyone is harassing or attempting to pickpocket you, try calling out "imdat!" (meaning “help!”) and contact the tourist police.
Visitors should also be aware that political demonstrations sometimes turn into violent confrontations between police and protestors, as in the Gezi Park (part of Taksim Square) riots of 2013. Galatasary Meydanı, on busy İstiklal Caddesi, is the starting point for many protests.