Lung and colon cancers are among the leading causes of human mortality and morbidity. Early diagnostic work up of these diseases include radiography, ultrasound, magnetic resonance imaging, and computed tomography. Certain blood tumor markers for carcinoma lung and colon also aid in the diagnosis. Despite the lab and diagnostic imaging, histopathology remains the gold standard, which provides cell-level images of tissue under examination. To read these images, a histopathologist spends a large amount of time. Furthermore, using conventional diagnostic methods involve high-end equipment as well. This leads to limited number of patients getting final diagnosis and early treatment. In addition, there are chances of inter-observer errors. In recent years, deep learning has shown promising results in the medical field. This has helped in early diagnosis and treatment according to severity of disease. With the help of models that have been cross-validated and tested fivefold, we propose an automated method for detecting lung (lung adenocarcinoma, lung benign, and lung squamous cell carcinoma) and colon (colon adenocarcinoma and colon benign) cancer subtypes from LC25000 histopathology images. A state-of-the-art deep learning architecture based on the principles of compound scaling and progressive learning, large, medium, and small models. An accuracy of 99.97%, AUC of 99.99%, F1-score of 99.97%, balanced accuracy of 99.97%, and Matthew's correlation coefficient of 99.96% were obtained on the test set using the -L model for the 5-class classification of lung and colon cancers, outperforming the existing methods. Using gradCAM, we created visual saliency maps to precisely locate the vital regions in the histopathology images from the test set where the models put more attention during cancer subtype predictions. This visual saliency maps may potentially assist pathologists to design better treatment strategies. Therefore, it is possible to use the proposed pipeline in clinical settings for fully automated lung and colon cancer detection from histopathology images with explainability.