The strobing is due to the way they’re powered. When converting AC power to DC power, you can either convert the positive half of the AC sine wave (half rectified) or both positive and negative sides of the AC wave (full rectified). Cheap lights use half rectifiers so as the AC feeds in, the light is only getting powered half the time and off the other half of the time. This happens so rapidly that we don’t really see it with our eyes, but with a camera it’s very noticeable. AC cycles at 60 cycles per second (in the US), so it makes sense that you’re seeing it at 120 shutter speed as this equates to 60fps.
I’d consider talking to some of these venues about it as I assume they’re typically used for events that’ll be filmed, so using shitty lighting is ruining it for everyone same as if they had the toilets sitting in the middle of the dance floor.