Human-robot cooperation is increasingly demanded in industrial applications. Many tasks require the robot to enhance the capabilities of humans. In this scenario, safety also plays an important role in avoiding any accident involving humans, robots, and the environment. With this aim, the paper proposes a cooperative fuzzy-impedance control with embedded safety rules to assist human operators in heavy industrial applications while manipulating unknown weight parts. The proposed methodology is composed by four main components: (i) an inner Cartesian impedance controller (to achieve the compliant robot behavior), (ii) an outer fuzzy controller (to provide the assistance to the human operator), (iii) embedded safety rules (to limit force/velocity during the human-robot interaction enhancing safety), and (iv) a neural network approach (to optimize the control parameters for the human-robot collaboration on the basis of the target indexes of assistance performance defined for this purpose). The main achieved result refers to the capability of the controller to deal with uncertain payloads while assisting and empowering the human operator, both embedding in the controller safety features at force and velocity levels and minimizing the proposed performance indexes. The effectiveness of the proposed approach is verified with a KUKA iiwa 14 R820 manipulator in an experimental procedure where human subjects evaluate the robot performance in a collaborative lifting task of a 10 kg part.