Tamil is an old Indian language with a large corpus of literature on palm leaves, and other constituents. Palm leaf manuscripts were a versatile medium for narrating medicines, literature, theatre, and other subjects. Because of the necessity for digitalization and transcription, recognizing the cursive characters found in palm leaf manuscripts remains an open problem. In this research, a unique Convolutional Neural Network (CNN) technique is utilized to train the characteristics of the palm leaf characters. By this training, CNN can classify the palm leaf characters significantly on training phase. Initially, a preprocessing technique to remove noise in the input image is done through morphological operations. Text Line Slicing segmentation scheme is used to segment the palm leaf characters. In feature processing, there are some major steps used in this study, which include text line spacing, spacing without obstacle, and spacing with an obstacle. Finally, the extracted cursive characters are given as input to the CNN technique for final classification. The experiments are carried out with collected cursive Tamil palm leaf manuscripts to validate the performance of the proposed CNN with existing deep learning techniques in terms of accuracy, precision, recall, etc. The results proved that the proposed network achieved 94% of accuracy, where existing ResNet achieved 88% of accuracy.