I will present some recent results on the coarse geometry of the Thurston metric on Teichmuller space. This is an asymmetric metric based on the Lipschitz constants of maps between hyperbolic surfaces. This metric is geodesic but geodesics are not unique and can behave quite wildly compared to Teichmuller geodesics. However, Thurston geodesics still behave nicely at the level of curve graphs of subsurfaces. Our main theorem is that, under suitable conditions, Thurston geodesics do not backtrack in subsurfaces. We can also give a characterization of when curves get short along a Thurston geodesic. A fundamental tool is the notion of "sufficiently horizontal" for a curve along a geodesic. This talk will represent joint work with David Dumas, Babak Modami, Anna Lenzhen, and Kasra Rafi.