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Bias or spurious correlations in image backgrounds can impact neural networks, causing shortcut learning (Clever Hans Effect) and hampering generalization to real-world data. ISNet, a recently introduced architecture, proposed the optimization of Layer-Wise Relevance Propagation (LRP, an explanation technique) heatmaps, to mitigate the influence of backgrounds on deep classifiers. However, ISNet's training time scales linearly with the number of classes in an application. Here, we propose reformulated architectures, dubbed Faster ISNets, whose training time becomes independent from this number. Additionally, we introduce a concise and model-agnostic LRP implementation, LRP-Flex, which can readily explain arbitrary DNN architectures, or convert them into Faster ISNets. We challenge the proposed architectures using synthetic background bias, and COVID-19 detection in chest X-rays, an application that commonly presents background bias. The networks hindered background attention and shortcut learning, surpassing multiple state-of-the-art models on out-of-distribution test datasets. Representing a potentially massive training speed improvement over ISNet, the proposed architectures introduce LRP optimization into a gamut of applications that the original ISNet model cannot feasibly handle.