Thulsa's bow is a different type from Subotai's. Seeing Thulsa's bow, it seems like he has what's called a recurve bow where, when unstrung, the limbs would cuve forward, historically very popular amongst nomadic horsemen. This was because, I believe, that it allowed them get a more powerful bow in a more compact form for use from horseback.
Subotai's bow, on the other, looks like a regular or straight bow where the limbs are more or less straight when unstrung.
I pulled up the movie tonight and I think Subotai's bow is an assymetrical bow!
THe top part is longer and the bottom arm is shorter with a more pronounced curve to it. It's a bit hard to see it in these screen caps but in motion it's much more obvious.
In the market scenes the bow is unstrung.
Here a few real mongolian and hun bows with a similar assymetrical structure-