A convenient way to get housing in Turku is through the Turku Student Village Foundation (TYS), which has around 400 apartments allocated to exchange students studying in a university in Turku for a semester or two. The application time for exchange students starts on May 1st for autumn semester and on October 1st for the spring. You can find their website and lots of important information here.
Other option is looking for housing from the private market. You can get started with, for example, the following websites:
When looking for an apartment from the private market or unofficial sources, remember to be careful and keep an eye out for scams, especially with offers on Facebook and similars, and keep in mind that you should never pay anything in advance.