They tend to make fun of ethnic minorities in Turkey. A lot of jokes are held at the Laz people's expense. The Laz are a group in the Black Sea region, and are widely stereotyped as stupid and weird.
As for rival countries, nearly every single country in the region can be considered Turkey's rival, especially the ones that were part of the Ottoman empire. A lot of these conflicts (I.e. with Bulgaria, Greece, and Armenia, or with the hypothetical "Kurdistan") are not exactly joking material and can get really heavy- not recommended at all.
If you need a target for humor, try the French or tourists in general.