The analytical expressions for the average intensity and angular spread of partially coherent Hermite-sinh-Gaussian (H-ShG) beams propagating through turbulent atmosphere are derived and used to study the evolution of average intensity distribution and angular spread of partially coherent H-ShG beams in turbulence. It is shown that an increase of the refraction index structure constant C2n and a decrease of the spatial correlation length σ0 accelerate the evolution process. The relative angular spread is introduced to quantitatively describe the resistance of a beam to turbulence. The smaller the spatial correlation length σ0, the waist width w0, the Sh-part parameter Ω0, and the larger the beam orders m, n, the less the angular spread of partially coherent H-ShG beams is affected by turbulence.