Hi Alex! Most if not all wood burning saunas in the center of Turku are private, so you want to look a little bit outside the immediate center of the city. Best public wood burning saunas are usually operated by winter swimming associations, so I'd recommend for instance a place by the name Saaronniemen Saukot which is located on the nearby island of Ruissalo. Despite being an island, the place is easily accessible by local public transportation (English language guide for the bus network here: https://reittiopas.foli.fi/Kauppatori%2C%20Kauppatori%20VI%2C%20Turku%3A%3A60.450564%2C22.266984/-/lähelläsi).

They have a modest website with some English language information here: http://www.saaronniemensaukot.fi/in-english

Another alternative is Villa Järvelä, also a winter swimming hotspot located by the small Lake Littoinen in the Littoinen area. Unfortunately they don't have a website in English, but you'll find the address in Google Maps. Entrance 5 EUR which includes three different saunas (2 of them wood burning) and a hot water outside bath.

Feel free to be in touch for more help and info!