We present a family of fully-discrete schemes for numerically solving nonlinear sub-diffusion equations, taking the weak regularity of the exact solutions into account. Using a novel discrete fractional Grönwall inequality, we obtain pointwise-in-time error estimates of the time-stepping methods. It is proved that as t→ 0 , the convergence orders can be σ, where σ is the regularity parameter. The initial convergence results are sharp. As t is far away from 0, the schemes give a better convergence results. Numerical experiments are given to confirm the theoretical results.