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This study introduces a computationally efficient convolutional neural network (CNN) architecture enhanced with transfer learning for multi-cancer detection using biomedical images. The proposed lightweight CNN model is designed to reduce computational complexity while maintaining high classification performance, making it suitable for deployment in resource-constrained environments. We evaluate this approach on three distinct tumor datasets comprising brain magnetic resonance imaging (MRI) and lung and kidney computed tomography (CT) scans. The model achieves test accuracy of 90.85 +- 2.22%, 98.64 +- 2.43% and 99.92 +- 0.08% for brain, lung, and kidney cancer classification, respectively, using 5-fold stratified cross-validation (CV). Transfer learning is employed by pretraining the model on one cancer type and fine-tuning it on the others, requiring only 20 additional epochs to achieve performance comparable to models trained from scratch. The fine-tuning process involves updating the classification part of the CNN and requires approximately 0.014 seconds per image per epoch using an NVIDIA GeForce GTX 960. Comparative evaluations show that the proposed model outperforms several state-of-the-art pretrained architectures, such as Xception, VGG16, VGG19, MobileNetV2 and DenseNet121. Overall, the model's effectiveness is evaluated across three types of cancer with distinct morphological characteristics, assessing its performance on both MRI and CT imaging modalities and demonstrating robust performance across diverse tasks and data types. These findings underscore the potential of streamlined deep learning (DL) frameworks in accelerating cancer diagnosis without sacrificing accuracy, especially in settings with limited computational resources.