He's linked to optoisolated relay boards. The simple solution is to use that capability.
With the jumper in place between VCC and JD-VCC you're powering both sides of the optoisolated circuits with the same supply. You need to remove that jumper and, with a separate 5V power source, power it on the JD-VCC and GND pins. Also, somewhat non-intuitively, remove any GND between the Arduino and the board. Leave a connection between the Arduino's VCC and the VCC on the relay board.