We study differentially private (DP) stochastic optimization (SO) with loss functions whose worst-case Lipschitz parameter over all data may be extremely large or infinite. To date, the vast majority of work on DP SO assumes that the loss is uniformly Lipschitz continuous (i.e. stochastic gradients are uniformly bounded) over data. While this assumption is convenient, it often leads to pessimistic risk bounds. In many practical problems, the worst-case (uniform) Lipschitz parameter of the loss over all data may be huge due to outliers and/or heavy-tailed data. In such cases, the risk bounds for DP SO, which scale with the worst-case Lipschitz parameter, are vacuous. To address these limitations, we provide improved risk bounds that do not depend on the uniform Lipschitz parameter. Following a recent line of work [WXDX20, KLZ22], we assume that stochastic gradients have bounded $k$-th order moments for some $k \geq 2$. Compared with works on uniformly Lipschitz DP SO, our risk bounds scale with the $k$-th moment instead of the uniform Lipschitz parameter of the loss, allowing for significantly faster rates in the presence of outliers and/or heavy-tailed data. For smooth convex loss functions, we provide linear-time algorithms with state-of-the-art excess risk. We complement our excess risk upper bounds with novel lower bounds. In certain parameter regimes, our linear-time excess risk bounds are minimax optimal. Second, we provide the first algorithm to handle non-smooth convex loss functions. To do so, we develop novel algorithmic and stability-based proof techniques, which we believe will be useful for future work in obtaining optimal excess risk. Finally, our work is the first to address non-convex non-uniformly Lipschitz loss functions satisfying the Proximal-PL inequality; this covers some practical machine learning models. Our Proximal-PL algorithm has near-optimal excess risk.