Marshrutkas / minibuses from Shymkent to Turkestan (Туркiстан) leave from the bus terminal next to the Bekzhan Market (or former market) and the massive Tsentralnaya Mosque at Map Location: 42.350153, 69.529224 . To get to the bus terminal take Marshrutka 47 or Bus 12 going north on Tamerlanovshoe Street for 75 tenge.
It's about a two hour ride to Turkestan and costs 1000 tenge. The minibuses leave when they are full and start loading at about 07:00 every morning. There are also big overnight buses at this station that go to Almaty and Astana.
In Turkestan, the minibus will turn right and then stop at the bus stop at Map Location: 43.299845, 68.276938 which is a few blocks from the Yasawi mausoleum complex. It is visible from the bus stop. The entry fee is 500 tenge.
To return to Shymkent, the marshrutkas leave from the Altyn Orda Bus Station at Map Location: 43.303553, 68.249294 .