In my experience when the fermented vegetable turns brown it is due to oxygen exposure or high temperature. It sounds like temperature was not the issue.
Did the recipe tell you to leave the jar uncovered? I've never heard anyone recommend doing that. With a lid on the jar the
CO2 will build up and will cause the ferment to bubble. With a lid on you must burp it regularly to allow these gases to escape, because the pressure will build. If you ferment without a lid then I would imagine that would prevent any noticable bubbling and would expose the vegetables to oxygen, even when submerged under a brine.
Also, 1/3 cup (6tbsp) of salt is way too much for 1 pound of turnips. This recipe reccomends only .5 tbsp per pound of turnips:
https://insaneinthebrine.com/sauerruben-fermented-turnips/