quarrel, dispute hotly
鬧 centers on heated, noisy conflict: a quarrel or a dispute that is argued passionately and loudly.
鬧 combines 鬥 (fighting, conflict) with 市 (market), suggesting the noisy, quarrelsome atmosphere of a busy marketplace. This is a standard semantic-component explanation.
Picture a fight (鬥) breaking out in a crowded market (市): people shouting and arguing loudly. That noisy quarrel is 鬧.
For トウ, imagine the loud quarrel reaching a peak when someone shouts 'TOW the line!' — the 'tow' sound cues トウ, and the argument is about rules.