Small LCD display help

You skipped checking if one of the TH pins is connected to the "-" pin (GND).
The point of this test is to see whether the board expects resistance or voltage as input.

Connect the ends of the pot to "-" and "+", connect the center to one of the TH pins.
It is better to not use the Arduino as a power suppy for this test, because if the TH connected to the pot is same as GND, turing the pot all the way will short + and GND and may damage the voltage regulator.