Glaucoma is a progressive optic neuropathy and one of the major causes of irreversible blindness worldwide. Early and accurate detection is essential to prevent vision loss, and retinal fundus imaging provides a non-invasive modality for clinical screening. In this paper, a customized convolutional neural network (CNN) is proposed for automated glaucoma classification using two publicly available datasets: Drishti-GS1 and ACRIMA. The proposed CNN consists of multiple convolutional and pooling layers with dropout regularization, designed to extract hierarchical features while avoiding overfitting. Data augmentation techniques such as rotation, shear, zoom, and brightness adjustment were applied to increase dataset diversity, and hyperparameters including batch size and learning rate were systematically tuned for optimal performance. Experimental results demonstrate that the proposed CNN achieved an accuracy of 90.32% on the Drishti-GS1 dataset and 96.45% on the ACRIMA dataset under an 80:20 split. The model also showed particularly high sensitivity, reaching 100% on Drishti-GS1 and 96.20% on ACRIMA, which is critical for minimizing false negatives in clinical screening. Furthermore, the proposed network outperformed widely used pre-trained CNNs such as AlexNet and ResNet-50, surpassing them in accuracy, sensitivity, and AUC. On the ACRIMA dataset, the model achieved an AUC exceeding 0.99, demonstrating its robustness and effectiveness as a reliable tool for automated glaucoma detection and screening.