Turnips are the first official vegetables that ever appear with their first appearance being in Yume Kojo: Doki Doki Panic. They are frequently found in Subcon, yet were originally found in the Real World.
Yume Kojo: Doki Doki Panic
Turnips only make an appearance in World 7-2 of Yume Kojo: Doki Doki Panic. They are produced from the Nightmare Machine, which the Arabian Family uses to throw vegetables at Wart's mouth to defeat him. The Turnips are the only effective items they could use as weapons against their foe to beat him. The bubbles Wart spits will fade way a Turnip if touched once.
Super Mario series
Super Mario Bros. 2
Turnips make their first Mario game appearance in Super Mario Bros. 2. They now have a different version than that from Yume Kojo: Doki Doki Panic. This time, Turnips are planted in the ground everywhere in Subcon, especially in grassy regions of the world. Turnips can be picked up and thrown at enemies to kill them. The Nightmare Machine can also produce Turnips from it, and they can be used as weapons for Mario to injure Wart, like in Yume Kojo: Doki Doki Panic.
BS Super Mario USA
Turnips make an appearance in the direct sequel, BS Super Mario USA, and are used for the same purpose as in the earlier game. Sometimes, enemies can be turned into these vegetables by certain characters' spells.
Super Smash Bros. Series
Princess Peach's down special move in Super Smash Bros. Melee and Super Smash Bros. Brawl is picking up turnips and using them as projectiles. The facial expression of the turnip determines the damage and knockback it causes (eg winking turnips cause more damage and knockback than normal smiling ones). The strongest variety of turnip, the sick turnip, is rare, but relatively powerful.
Captain Toad: Treasure Tracker