Sufi orders are officially banned in Turkey, which isn't to say that they don't exist there, but that they are highly politicised and mostly exist underground. You are unlikely to find them as a tourist.
The modern Mevlevî "whirling dervish" performances are specially designed for tourists; they might be visually authentic but they do not represent a functioning sufi order. The Mevlevî museum in Konya is pleasant enough to visit, but there isn't a whole lot going on there.
One lesser-visited site in Turkey with connections to sufism is the shrine to the mystic Hacı Bektaş Veli in the town of Hacıbektaş in the province of Nevşehir (close to Cappadocia). This is not a functional sufi lodge, but Turkey's largest religious minority, the Alevis, regard it as a holy site, so it is much more relevant than any of the "touristic sufi" locations in Turkey. I believe they hold performances of some sort there at certain times of year and there is an annual pilgrimage (in December?).
You can also visit sufi shrines in many other countries, but outwardly most are not much different than any other burial places for holy men. For example, there's a large complex for the founder of the Naqshbandi sufi order just outside Bukhara in Uzbekistan. The only place I've ever witnessed an actual sufi devotional ceremony was in Pakistan, but that doesn't fit the criteria you've set out in your question.