The learning objective of vision-language approach of CLIP does not effectively account for the noisy many-to-many correspondences found in web-harvested image captioning datasets, which contributes to its compute and data inefficiency. To address this challenge, we introduce a novel training framework based on cross-modal contrastive learning that uses progressive self-distillation and soft image-text alignments to more efficiently learn robust representations from noisy data. Our model distills its own knowledge to dynamically generate soft-alignment targets for a subset of images and captions in every minibatch, which are then used to update its parameters. Extensive evaluation across 14 benchmark datasets shows that our method consistently outperforms its CLIP counterpart in multiple settings, including: (a) zero-shot classification, (b) linear probe transfer, and (c) image-text retrieval, without incurring added computational cost. Analysis using an ImageNet-based robustness test-bed reveals that our method offers better effective robustness to natural distribution shifts compared to both ImageNet-trained models and CLIP itself. Lastly, pretraining with datasets spanning two orders of magnitude in size shows that our improvements over CLIP tend to scale with number of training examples.