In this paper, we study nonsmooth composite optimization problems under heavy-tailed noise, with the objective being a summation of a nested function and a nonsmooth convex regularizer. We propose stochastic proximal approximation methods incorporating a normalization technique to handle the potential challenges caused by the nonsmooth regularizer and heavy-tailed noise. For the case where the outer function of the nested structure is smooth, our proposed algorithms achieve sample complexities that match the best known results for single-layer nonconvex stochastic optimization under heavy-tailed noise. For the case where the outer function is convex but nonsmooth, to the best of our knowledge, the corresponding normalized stochastic proximal gradient methods are new, with sample complexity bounds provided. For each of the above proposed algorithms, we consider two variants with constant parameter sequences and decaying ones, respectively. The effectiveness of the proposed methods is validated through numerical experiments on sparse phase retrieval problem and policy evaluation for Markov decision processes.