As Rose pointed out, the minimum number of turns per stage is actually 0. This is done in two main ways, that I know of: Retaliation gear, and relying on good RNG with a slow champion with AoE on their A1 (like Skullcrown and Sinesha).
The other is letting your champion(s) being constantly hard CC-ed as much as possible, which is what I do on Scarab. When they are provoked, for example, they are technically not taking a turn since they have no choice in what they do (they are literally being controlled). Therefore, the game doesn't count those "turns"as turns.
One other example is Nether Spider. The lowest turn count is done with Geomancer being the fastest and putting his HP burn on the mother spider. He is followed by someone to put Heal Reduction on the main spider. The other champions are A1 AoE champions that then attack everyone, spiderlings counterattack - but the game treats spiderlings as part of the main spider - so Geo's passive deals massive damage to the main spider every time the spiderlings counterattack.
As to why we do it - bragging rights, of course! What else?😉