Myoelectric hand prostheses offer a way for upper-limb amputees to recover gesture and prehensile abilities to ease rehabilitation and daily life activities. However, studies with prosthesis users found that a lack of intuitiveness and ease-of-use in the human-machine control interface are among the main driving factors in the low user acceptance of these devices. This paper proposes a highly intuitive, responsive and reliable real-time myoelectric hand prosthesis control strategy with an emphasis on the demonstration and report of real-time evaluation metrics. The presented solution leverages surface high-density electromyography (HD-EMG) and a convolutional neural network (CNN) to adapt itself to each unique user and his/her specific voluntary muscle contraction patterns. Furthermore, a transfer learning approach is presented to drastically reduce the training time and allow for easy installation and calibration processes. The CNN-based gesture recognition system was evaluated in real-time with a group of 12 able-bodied users. A real-time test for 6 classes/grip modes resulted in mean and median positive predictive values (PPV) of 93.43% and 100%, respectively. Each gesture state is instantly accessible from any other state, with no mode switching required for increased responsiveness and natural seamless control. The system is able to output a correct prediction within less than 116 ms latency. 100% PPV has been attained in many trials and is realistically achievable consistently with user practice and/or employing a thresholded majority vote inference. Using transfer learning, these results are achievable after a sensor installation, data recording and network training/fine-tuning routine taking less than 10 min to complete, a reduction of 89.4% in the setup time of the traditional, non-transfer learning approach.