You should try to make a schematic of your board if you want to learn why it is behaving that way.
Here is what I
believe you have (borrowing these from
https://www.handsontec.com/dataspecs/4Ch-relay.pdf) but I think they are basically representative.
It probably only takes about 5mA or so to operate the optoisolator. In other words, that is all the GPIO needs to supply. The problem is, where is the power to operate the relay coil going to come from? It has no choice but to come from the module VCC and therein lies your situation. If the PS that feeds the ESP8266 has enough left over, then that will work but you have to have GND connected also.
Other boards will have a jumper on them, like this:
In that case, you could have an external PS with an external GND.
You do not have that option. You don't have isolation other than the relay.
They, apparently, decided that they would leave that jumper out. If you make your own schematic, you might be able to modify it. Alternatively, get a better board.
I could be missing something, but that is how I see it.