The application of deep learning in medical imaging has significantly advanced diagnostic capabilities, enhancing both accuracy and efficiency. Despite these benefits, the lack of transparency in these AI models, often termed "black boxes," raises concerns about their reliability in clinical settings. Explainable AI (XAI) aims to mitigate these concerns by developing methods that make AI decisions understandable and trustworthy. In this study, we propose Tumor Aware Counterfactual Explanations (TACE), a framework designed to generate reliable counterfactual explanations for medical images. Unlike existing methods, TACE focuses on modifying tumor-specific features without altering the overall organ structure, ensuring the faithfulness of the counterfactuals. We achieve this by including an additional step in the generation process which allows to modify only the region of interest (ROI), thus yielding more reliable counterfactuals as the rest of the organ remains unchanged. We evaluate our method on mammography images and brain MRI. We find that our method far exceeds existing state-of-the-art techniques in quality, faithfulness, and generation speed of counterfactuals. Indeed, more faithful explanations lead to a significant improvement in classification success rates, with a 10.69% increase for breast cancer and a 98.02% increase for brain tumors. The code of our work is available at https://github.com/ispamm/TACE.