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Before You Go
You won't need special inoculations before entering Turkey unless you're coming from an endemic or epidemic area. However, do discuss your requirements with a doctor. Consider typhoid fever and hepatitis A and B vaccinations if you plan to travel off the beaten track in Turkey. Also make sure that your tetanus/diphtheria and polio vaccinations are up to date (boosters are necessary every 10 years).
Food & Water
Standards of food hygiene are generally high in İstanbul and visitors experience few food-related illnesses. To be safe, treat street food with caution and if you dine in a lokanta (restaurant serving ready-made food) make sure you choose dishes that look hot and freshly prepared.
Tap water in İstanbul is chlorinated, but is still not guaranteed to be safe (many locals don't drink it). Spring water is cheap and sold everywhere in 330mL, 1.5L and 3L plastic bottles.
Turkey doesn't have reciprocal health-care arrangements with other countries, so having travel insurance is highly advisable.
For minor problems it's customary to ask at a chemist/pharmacy (eczane) for advice. Many pharmacists speak English and will prescribe treatment on the spot. Drugs requiring a prescription in Western countries are often sold over the counter (except for the most dangerous or addictive ones) and will often be cheaper too. Make sure that you know the generic name of your medicine; the commercial name may not be the same in Turkey.
Most doctors in Turkey speak English and half of all the physicians in İstanbul are women. If a woman visits a male doctor, it's customary for her to have a companion present during any physical examination or treatment.
Though they are expensive, it's probably easiest to visit one of the private hospitals listed here if you need medical care when in İstanbul. Their standard of care is generally quite high and you will have no trouble finding staff who speak English. Both accept credit-card payments and will charge around ₺300 for a standard consultation.
American Hospital Pediatric, dental and many other clinics.
Memorial Sişli Hospital Emergency department, eye centre and pediatric clinic.