"Took out a Big Arms car using only melee"
You can ONLY use melee attacks (X or Y) to damage the enemy, if you accidentally shoot them or ram them with turbo, move on to a different car.
Just try boosting yourself slightly ahead of them, stick to a side far away so you don't accidentally ram them. Once you get ahead, move towards them and start mashing Y to hit with Pablo. Should kill them in a hit or two, especially if fully upgraded.
Easiest done in level 1-1 or 1-2, as the enemies come in small groups and usually stay spread out enough for you to pass them.