Bad earth to light cluster and "something" is pulling the cluster "earth" up towards 12V ? When you switch lights on, the lamp goes out because it's now between 12V and 12V.
Probably not full on and off - more likely dim and dimmer.
You see it a fair bit where a tail light is on but goes off when either brake or indicator is used - with brake/indicator light not coming on.
So first check, measure voltage between earth on the light and chassis.