This city is one of the most conservative cities in Turkey and has a reputation for it. Many non-local Turks will swear that you will not even have a slight chance for a lift around Konya, but that’s not true. Instead, this is one of the places in Turkey where you will be offered quickest lifts, even if you don’t have a conservative appearance according to local standards (e.g. a male having long hair and an untidy beard).
Get yourself out of town by either catching a bus in direction Hayvanat Bahcesi, or try to hitchhike. It's easy to find the way out, everywhere are signs that show you the way towards Adana. At the beginning of the Eregli-Adana karayolu , or the road D330, you find a petrol station, and behind of it enough space to stop cars. Keep in mind that for the next 100 km or so, no bigger villages will pass your way, so best case is to catch a vehicle that goes directly towards Eregli or even further.
Catch the bus towards Karaman Yolu (i.e. “Road to Karaman”, bus #26 goes there, and in April, 2008 costed YTL 1.10/person, one-way, which is about half a Euro). You can catch it from the stop in the city centre, in front of the Governer’s Office (Valilik or Vilayet), which is not very far from the Mevlana Museum/Rumi tomb or Alaaddin Park. This bus will take you fairly out of city, about 10-15 km, on the motorway D715. Just push the stop button as soon as it quits the higway by turning left into a much narrower road. After getting off the bus, cross the motorway back to the other side (it’s not very hard) and start thumbing. You won’t wait long (~15-20 minutes), but as the cars will be passing by quite fast, always watch your back to see if they had stopped tens of metres ahead. Be aware that this spot is almost in the middle of nowhere and there are hardly any buildings around (let alone a shop), so you may consider taking some extra water before you board the bus: It’s the steppes after all, and it’s literally baking under the sun during summer. Also another thing to consider is that you will pass through only three towns of considerable size (Iceri Cumra (pronounced ee-cheh-ree choom-raa), Karaman, and Mut respectively) all along the way down to Silifke, and for some hundred(s) of kilometres (between Iceri Cumra and Karaman) not a single village along the way, so don’t accept lifts to other than these or directly Silifke itself. Getting a lift right into Karaman city centre will make you stray from the motorway about a few kilometres away, so also take this into consideration.
If you don't have a place to sleep in Konya, you could also go to the central bus station in the city center. It seems to be open 24 hours a day. Platschi arrived there at 4 a.m. in the night and found enough benches to lay down. Many other local people also did, waiting for their bus to arrive. Also security was walking around there. A very good place for wildcamp is the suburban park in the western part of the city (Meram belediyesi dutlu kulturpark) where a mosque, free public toilet (almost unique in Turkey..) and several bars on the edges of a dirty sewer river make the perfect environment for a nice stay. Fede stayed there 2 nights leaving also his backpack at one of the bars and getting free hot water for cook and tea all the time he needed. The central park on the hill Alaaddin tepesi seems to be less safe and calm listening to locals.