The counry is changing. However, it is important to know that being transgender is not new or a western idea. The Najajo and many other native groups, Thai, and Indian people and many others have third genders or transgender people of some kind in their culture. Accepting them is new to the Western world.
Yes, Turkey is on the verge of embracing modern lifestyles. People in the country feel that those who think and act differently should be embraced within the country. This can be seen as a form of progress within the country of Turkey. Modern lifestyles are becoming more and more acceptable there.
Turkey is a secular state resting in the midst of nations heavily influenced by religious laws and beliefs, and it has the opportunity to move the modernization of social norms along in the Middle East region. Protests are happening on an individual level and a collective level calling for greater social rights for individuals and the trend is accelerating in this region of the world.
It was nice to see that a lot of Turkish people took to the streets to protest the murder of a transgender woman, but the majority of Turks and their government have a long way to go before we can say they have embraced a modern lifestyle. It took decades in the US, it will likely take longer there.