you can run a jump line from your headlight line (brights)to your fog power, as of right now it just is plugged into your headlight (regular light) that is why when you flip the brights it goes off, a bridge would allow the power to flow

 

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Doing that will short the seperate high and low beam circuits together causing both the high & low beam filaments of the headlamp to come on at the same time, not only creating very high heat in the headlamp housing but blowing the Lights fuse in short order.