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We explore the potential of CNN-based models for gallbladder cancer (GBC) detection from ultrasound (USG) images as no prior study is known. USG is the most common diagnostic modality for GB diseases due to its low cost and accessibility. However, USG images are challenging to analyze due to low image quality, noise, and varying viewpoints due to the handheld nature of the sensor. Our exhaustive study of state-of-the-art (SOTA) image classification techniques for the problem reveals that they often fail to learn the salient GB region due to the presence of shadows in the USG images. SOTA object detection techniques also achieve low accuracy because of spurious textures due to noise or adjacent organs. We propose GBCNet to tackle the challenges in our problem. GBCNet first extracts the regions of interest (ROIs) by detecting the GB (and not the cancer), and then uses a new multi-scale, second-order pooling architecture specializing in classifying GBC. To effectively handle spurious textures, we propose a curriculum inspired by human visual acuity, which reduces the texture biases in GBCNet. Experimental results demonstrate that GBCNet significantly outperforms SOTA CNN models, as well as the expert radiologists. Our technical innovations are generic to other USG image analysis tasks as well. Hence, as a validation, we also show the efficacy of GBCNet in detecting breast cancer from USG images. Project page with source code, trained models, and data is available at https://gbc-iitd.github.io/gbcnet