Hi Zedax,
Power supply for SerialLCD should be from power conectors - oposite side to pins 10,11.
On “power side” you have pins: reset, 3V3, 5V, GND, GND, Vin, then in next segment analog pins 0 to 5.
You should take GND and 5V to supply Serial LCD and 10,11 pins as signal TX,RX - it work fine for me.
SerialLCD module require +5V supply. When connected to +3.3 - control leds on module light propper, but module not working propper.
It is also possible to use external power source +5V, bu in this case you should connect both GND - from Arduino board and from your external power source, in other case - RX and TX signals on SerialLCD have no GND connection to Arduino.
Best,
Marek