To propose my own theory for the development of the kilic, I think it came from the Mongols, through the Persians, and then on to the Ottomans and Mamluks. Here are a couple of miniatures which support this view.
14th century Mongol painting. The man's sword is a bit tough to see, but note the yelman:
14th century Persian battle scene. Note the very bottom of the image, there is a curved sword with a yelman:
Persian battle scene from 1314. Note the curved sword, but without a yelman this time:
