We consider network security exercises where students construct virtual networks with User-mode Linux (UML) virtual machines and then execute attack and defense activities on these networks. In an older version of the exercise system, the students accessed the desktop screens of the remote servers running UMLs with Windows applications and then built networks by executing UML commands. However, performing the exercises remotely (e.g., due to the COVID-19 pandemic) resulted in difficulties due to factors such as the dependency of the work environment on specific operating systems, narrow-band networks, as well as issues in providing support for configuring UMLs. In this paper, a novel web-based hands-on system with intuitive and seamless operability and lightweight responsiveness is proposed in order to allow performing the considered exercises while avoiding the mentioned shortcomings. The system provides web pages for editing device layouts and cable connections by mouse operations intuitively, web pages connecting to UML terminals, and web pages for operating X clients running on UMLs. We carried out experiments for evaluating the proposed system on the usability, system performance, and quality of experience. The subjects offered positive assessments on the operability and no negative assessments on the responsiveness. As for command inputs in terminals, the response time was shorter and the traffic was much smaller in comparison with the older system. Furthermore, the exercises using nano required at least 16 kbps bandwidth and ones using wireshark required at least 2048 kbps bandwidth.