Although automated Skin Lesion Classification (SLC) is a crucial integral step in computer-aided diagnosis, it remains a challenging task due to inconsistency in texture, color, indistinguishable boundaries, and shapes. In this article, we propose an automatic and robust framework for the dermoscopic SLC named Dermoscopic Expert (DermoExpert). The DermoExpert consists of a preprocessing, a hybrid Convolutional Neural Network (hybrid-CNN), and transfer learning. The proposed hybrid-CNN classifier consists of three distinct feature extractors, with the same input images, which are fused to achieve better-depth feature maps of the corresponding lesion. Those distinct and fused feature maps are classified using the different fully connected layers, which are then ensembled to get a final prediction probability. In the preprocessing, we use lesion segmentation, augmentation, and class rebalancing. For boosting the lesion recognition, we have also employed geometric and intensity-based augmentation as well as the class rebalancing by penalizing the loss of the majority class and adding extra images to the minority classes. Additionally, we leverage the knowledge from a pre-trained model, also known as transfer learning, to build a generic classifier, although small datasets are being used. In the end, we design and implement a web application by deploying the weights of our DermoExpert for automatic lesion recognition. We evaluate our DermoExpert on the ISIC-2016, ISIC-2017, and ISIC-2018 datasets, where our DermoExpert has achieved the area under the receiver operating characteristic curve (AUC) of 0.96, 0.95, and 0.97, respectively. The experimental results outperform the recent state-of-the-art by a margin of 10.0 % and 2.0 % respectively for ISIC-2016 and ISIC-2017 datasets in terms of AUC. The DermoExpert also outperforms, in concerning a balanced accuracy, by a margin of 3.0 % for ISIC-2018 dataset. Since our framework can provide better-classification on three different test datasets, even with limited training data, it can lead to better-recognition of melanoma to aid dermatologists. Our source code, and segmented masks, for ISIC-2018 dataset, will be made publicly available for the research community for further improvements.