1

I've been to Turkey with my wife three times in the last 10 years, traveling the Med, Aegean, Black Sea, and Marmara coasts, and a lot of the interior. As budget travelers, we often share accommodation and transportation with singles of both sexes, because sharing costs makes such good sense, and we often meet up with those same people weeks after our shared experiences, and exchange stories. We've never in a cumulative total of 4-5 months in Turkey met a single female traveler who had encountered anything heavier than an easily-deflected pass.
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On buses, there is a strict protocol. In fact, no man should sit beside a woman who isn't his wife or a relative. I was asked to move on a crowded bus because a single young female got on, and the only seat available was next to a young man. The conductor asked me to take that seat, so the single girl could sit beside my wife. We both had interesting conversations with our new seatmates as a result.
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Sarikanarya has given you good advice. Call me a chauvinist, but I've always felt it was my duty to protect women from violence or unwanted advances. Over the last few years, though, I have become much more circumspect in the increasingly frequent instances where the woman is drinking, flirting, and flaunting. If a woman dragged a bloody steak in front of a hungry dog without worrying about the consequences, I'd be disinclined to leap eagerly to her defense as well. I might reluctantly write her problem off to natural selection, even though it's my instinct to be protective.
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Turkish women in general and especially in Western Turkey are proud and assertive people. They aren't timid medieval maidens hiding behind a veil, or in purdah. The conservatism of Turkish society is meant to protect their honor, and it will protect yours equally if you conduct yourself with dignity and common sense. Behave like a chaste woman, and your chastity and attitude will both be respected.
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Of course, hearing of your unfortunate experience in a London backpackers, I've now decided that it's not safe to take my wife to England.
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Not really. Enjoy Turkey. Cirali is nicer than Olympos, IMHO.

2

The warnings seem directed towards you by those who see and know you.

Young, pretty, friendly, made up, coy, presumed sexually active single female.

Leave most of the makeup at home, cover your bosom, shoulders and knees in public. Behave primly and properly. Limit your interacti/on with men to the conduct of business when no female is available to assist you and do not go out alone after dark. Such behavior is an effective guarantee against problems.

Understand that in touristy coastal areas young Turkish gigolos are far more common than Turkish prostitutes so don't be too disappointed if you go clubbing and it is you who is expected to pay.

3

Great advice from all the above. After having lived in Istanbul for a while we travelled to London for a few days. When on the subway I felt a little less safe than when I first went into a 1960's New York City subway... Which part of London do you NOT want to be caught in at night?

You got a full load of rubbish (crap in the USA) from whomever you received your "warnings." Thousands of solo women travellers have been coming to Turkey for years. Those that get into trouble are usually those who THINK they are looking for a "Turkish delight." I used to get such warnings from my university colleagues in Istanbul about walking through certain neighborhoods on the way to work. Again, after 7 years of such activity, day and night walking, I never heard so much as a boo...

You did say you are travelling alone, correct? Not with your daughter? Makes no difference, same rules apply...

Watch what you do and listen to the "uh-oh" inside of you and you should be just fine.
Take a look through the following:
(sarıkanarya already gave you the link to her posting)

Women Traveling in Turkey by Tom Brosnahan of Turkey Travel Planner

Natalie Sayin, former Brit and currently married to a Turk says in her article Turkish Men – Why Your Holiday Romance Is Doomed, "Do not get involved in a holiday romance with a Turkish man simply because you can not handle it."

Also see her companion pieces; Is it safe to travel to Turkey? AND Travel Myths about Turkey Busted

And if by now you need more encouragement read Seven Solo Female Travel Myths Debunked !

A Lonely Planet thread called Why I Keep Returning to Turkey (and why more from the USA should visit)

Are Americans safer in Mexico than at home? by Robert Reid Lonely Planet author
"Every week or so I get asked, ‘Is it safe to go to Mexico?’ I had always said, if you’re thoughtful about where you go, yes. But after my most recent trip there, I’m changing my answer… to a question: Do you think it’s safe to go to Texas?" Mr Reid writes about Mexico but I hear the same stuff about Turkey.

You might want to avoid these:

Come Back Alive - A list of countries you probably should avoid; Turkey is NOT on this list.

The World's Most Dangerous Places
http://en.wikipedia.org/wiki/The_World's_Most_Dangerous_Places
"...is a frequently updated handbook written by Pelton and an unusual crew of contributors."

America's Most Dangerous Cities
http://www.forbes.com/2009/04/23/most-dangerous-cities-lifestyle-real-estate-dangerous-american-cities.html

4

Relax, you'll be fine. Use normal precautions that you would use when traveling anywhere in the world, but Turkey is likely to be safer than your home country.

5

Turkey is not more dangerous than any country. But you need to approach with prudence to the people even if they look very friendly and avoid the risks.
For example, if you hitchhiking of course it can be dangerous. The other users already gave good advices.
The safety rules are almost same in any country. If you apply the simple rules then your trip will be really safe.

6

I visited Turkey last year with my husband and my mother (80 yrs old). I understand it is not exactly the same situation of yours. However, I did not feel unsafe anywhere. Of course, I was always accompanied by my husband, and he does look like a local, though he is Brazilian. I think you should take good care and consider all the advice given above, that really refer to any other place in the world! When my mother was together, we hired a car to make things easier for her, but as previously arranged, from the Capadocia part of the visit she would return home. So, when we were on our own, we used public transport most of the times.We took a bus at night in Ankara to go Çankaya district, travelled by bus from Ankara to Bolu, and even took a lift in Bolu. No trouble whatsoever. Good luck. If you´re interested in my report of this trip, please see my blog at http://curiosidadesdeviagens.blogspot.com.br/.