Though recent works have focused on modeling high resource languages, the area is still unexplored for low resource languages like Bengali and Hindi. We propose an end-to-end trainable memory efficient CNN architecture named CoCNN to handle specific characteristics such as high inflection, morphological richness, flexible word order and phonetical spelling errors of Bengali and Hindi. In particular, we introduce two learnable convolutional sub-models at word and at sentence level that are end-to-end trainable. We show that state-of-the-art (SOTA) Transformer models including pretrained BERT do not necessarily yield the best performance for Bengali and Hindi. CoCNN outperforms pretrained BERT with 16X less parameters and achieves much better performance than SOTA LSTMs on multiple real-world datasets. This is the first study on the effectiveness of different architectures from Convolution, Recurrent, and Transformer neural net paradigm for modeling Bengali and Hindi.