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.
TREE, KING, WATCH, and TOOTH all fit the category "Things with crowns". A crown can sit on each of these. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Things that fly
KITE / JET / EAGLE / DRONE
KITE, JET, EAGLE, and DRONE all fit the category "Things that fly". They spend time in the air. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Film crew roles
PRODUCER / DIRECTOR / EDITOR / GAFFER
PRODUCER, DIRECTOR, EDITOR, and GAFFER all fit the category "Film crew roles". They appear in movie credits. 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
BENCH / KETTLEBELL / ROPE / MAT
BENCH, KETTLEBELL, ROPE, and MAT 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.