The full solution is below, with a short explanation for each group. To avoid spoilers, play the puzzle or open the hints page before reading the answer panels.
ROAD, TRACK, BOWLING, and POOL all fit the category "Things with lanes". Movement is organized into lanes. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Desert words
CACTUS / OASIS / DUNE / MIRAGE
CACTUS, OASIS, DUNE, and MIRAGE all fit the category "Desert words". Think sand and heat. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Words after moon
BEAM / LIGHT / STONE / WALK
BEAM, LIGHT, STONE, and WALK all fit the category "Words after moon". Put one word before each answer. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Things in a gym
ROPE / MAT / BENCH / KETTLEBELL
ROPE, MAT, BENCH, and KETTLEBELL all fit the category "Things in a gym". Training rooms are their habitat. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.