Brain tumors (BTs) are spreading very rapidly across the world. Every year, thousands of people die due to deadly brain tumors. Therefore, accurate detection and classification are essential in the treatment of brain tumors. Numerous research techniques have been introduced for BT detection as well as classification based on traditional machine learning (ML) and deep learning (DL). The traditional ML classifiers require hand-crafted features, which is very time-consuming. On the contrary, DL is very robust in feature extraction and has recently been widely used for classification and detection purposes. Therefore, in this work, we propose a hybrid deep learning model called DeepTumorNet for three types of brain tumors (BTs)—glioma, meningioma, and pituitary tumor classification—by adopting a basic convolutional neural network (CNN) architecture. The GoogLeNet architecture of the CNN model was used as a base. While developing the hybrid DeepTumorNet approach, the last 5 layers of GoogLeNet were removed, and 15 new layers were added instead of these 5 layers. Furthermore, we also utilized a leaky ReLU activation function in the feature map to increase the expressiveness of the model. The proposed model was tested on a publicly available research dataset for evaluation purposes, and it obtained 99.67% accuracy, 99.6% precision, 100% recall, and a 99.66% F1-score. The proposed methodology obtained the highest accuracy compared with the state-of-the-art classification results obtained with Alex net, Resnet50, darknet53, Shufflenet, GoogLeNet, SqueezeNet, ResNet101, Exception Net, and MobileNetv2. The proposed model showed its superiority over the existing models for BT classification from the MRI images.