Currently, Coronavirus disease (COVID-19), one of the most infectious diseases in the 21st century, is diagnosed using RT-PCR testing, CT scans and/or Chest X-Ray (CXR) images. CT (Computed Tomography) scanners and RT-PCR testing are not available in most medical centers and hence in many cases CXR images become the most time/cost effective tool for assisting clinicians in making decisions. Deep learning neural networks have a great potential for building COVID-19 triage systems and detecting COVID-19 patients, especially patients with low severity. Unfortunately, current databases do not allow building such systems as they are highly heterogeneous and biased towards severe cases. This article is threefold: (i) we demystify the high sensitivities achieved by most recent COVID-19 classification models, (ii) under a close collaboration with Hospital Universitario Clínico San Cecilio, Granada, Spain, we built COVIDGR-1.0, a homogeneous and balanced database that includes all levelsof severity, from normal with Positive RT-PCR, Mild, Moderate to Severe. COVIDGR-1.0 contains 426 positive and 426 negative PA (PosteroAnterior) CXR views and (iii) we propose COVID Smart Data based Network (COVID-SDNet) methodology for improving the generalization capacity of COVID-classification models. Our approach reaches good and stable results with an accuracy of 97.72% ± 0.95%, 86.90% ± 3.20%, 61.80% ± 5.49% in severe, moderate and mild COVID-19 severity levels. Our approach could help in the early detection of COVID-19. COVIDGR-1.0 along with the severity level labels are available to the scientific community through this link https://dasci.es/es/transferencia/ open-data/covidgr/.