How do I deal with working out with colored hair?

I workout almost everyday and the roots of my hair get wet from all the sweat and it’s worse when I wear a hat the hair dye strips off even faster and I don’t know what to do to keep my hair color. I wash my hair at least once a week and even at that a lot of hair dye comes off. I don’t want to stop working out to keep my hair color but I can’t keep dying it often since I’ve started losing a lot of hair too. I use the $4 box dye to be fair but if I purchased a better quality brand wouldn’t the same thing keep happening?