By and large, Istanbul is a very safe city. Like all large cities, normal precautions are required and just as you wouldn't take any chances in New York or London, don't in Istanbul. The four most common threats to personal safety in Istanbul are all associated with your money. The first of these threats is purse-snatching, a global game everywhere one goes. Purse -snatching tends to occur in tourist areas (where you'll be spending all your time in Istanbul), and is generally done by children just approaching their teens. It's best not to chase down the scamps, even if you are able, as most certainly their parental guardians are nearby, and this is a confrontation best avoided.

The second thing to watch out for is the universal taxi driver scam. Your taxi driver will not speak English, and the Turkish equivalent of even "Yes" and "No" is so completely unintelligible that even basic communication is a struggle. Your hotel will give you a card with the name and address so you can get back to where you started. However, the trip can be a long one, the price usurious, and you can be certain the taxi driver will have no change. Always have lots of change. There is a difference between the day rate and the night rate on the meter. Be very certain that as a foreigner, you will be charged the night rate, which is somewhat more, as soon as you enter the taxi. This is not worth arguing about.

 

People 

Turkish hospitality is justifiably famous, though sometimes with a twist. You may meet some very friendly people who will invite you for refreshments at a nearby bar or restaurant. After a few drinks, you may be presented with a very large bill, often amounting to several hundred dollars. Claiming poverty, you will then be escorted to the nearest bank machine to withdraw the necessary funds, amid some threatening noises. This kind of thing typically is confined to the Taksim or Laleli area, especially  male-only 'night clubs' with presence of  women from Eastern Europe.

A gentler but still insidious version is where someone - perhaps even introduced to you by a Turk you know -  attaches himself socially then tries to sell you something, like a carpet. Since you are now "friends" and he has perhaps spent several hours reeling you in, including perhaps an hour or two viewing carpets in his (or his associates') premises (perhaps as part of a personal guided tour of the Bazaar), you have to be firm.

If you are a single woman or a woman-only group, you should be a bit more cautious when someone or  a group of men try to approach you or are following you. Needless to say the same is true if you are in a bar. Turkish men also might interpret an innocent act very differently. 

Simple precautions 

if you have a bag first, check once a while if it is open and second,  carry it  on your shoulder or hang it across your chest. Hold your cell phone while speaking, as cell phones are the main target of street theft and it can happen anywhere in Istanbul anytime of the day. And don't have a lot of money with you as there are cash-machines everywhere.

Daytime

If you decided to discover others part of the city where not many tourists or even locals are visible, just keep in mind that virtually all neighbourhoods can be considered safe even the slums or poor ones.  There is no real danger of murder. However, the best rule is to walk where you see a lot of people esp. women. Each neighbourhood normally has a main street where the grocery stores are open until late in the evening (21-22 PM at least). Neighbourhoods like esp. Tarlabasi, Dolapdere and some parts of Galata and Karakoy around are better to be avoided even during daytime. If you decide to take a taxi and if you can, prefer  to pick one from a taxi station and not one just passing by especially at night. 

Istanbul at Night

Despite improvements, most parts of the city is quite dark and empty after sunset. There is also a big difference between winter and summer when people stay late and go out much often. As criminality is on rise, try to avoid dark or suspicious sidestreets, especially outside Sultanahmet in the old city and in poorer neighbourhoods. If you want to go to some specific restaurant, club or cafe, prefer to take a bus,taxi or dolmus. Most of the bus lines don't function after 10 PM. Walk at least two people together, preferably with at least one male present and don't take much money with you, because you can always pay by credit card or use the cash-machines. 

Until midnight, the safest  parts are Sultanahmet, Beyoglu (Pera), Taksim and European part of Bosphorus. Middle class neighbourhood centers of  Kadikoy, Uskudar, main roads on the Asian side and Taksim-Levent axis (where the subway runs) on the European side are also fine.

Traffic 

Finally, and for most visitors to Istanbul, the most dangerous thing you will do is simply crossing the street. Traffic is largely punishing gridlock day and night, with little of what passes as traffic regulations elsewhere. Be alert and cautious. Just because the light says cross, doesn't neccesarily mean it is safe to do so.

Medicines, doctors and medical care

Istanbul has some very good private hospitals, an excellent ambulance service, and good medical care. What you will not find in Istanbul is a drug store experience, or even drug store over the counter medecines that you are used to finding at home. These simply do not exist in Turkey. Your hotel will be able to assist you, but it's always best to bring your own.