We need to first separate the three kinds of two-turn moves before answering this question.
- Those that give turns of semi-invulnerability, which includes moves like Dig and Fly.
- Those that require a turn of charging before activation, which includes moves like Geomancy and Meteor Beam.
- Those that need a turn to recharge after being used, which includes moves like Hyper Beam and Giga Impact.
The biggest reason for all of them is that it's simply too easy for the opponent to predict or take advantage of the Pokemon using them. If you use Dig, for example, the opponent has a free switch-in to their Flying type, can Protect to block all damage, or even start racking up stat boosts and hazards if they can afford to stay in. Two turn moves usually sacrifice too much momentum for them to be worth it.
The other big reason to not use semi-invulnerable moves is that most of the time, a Pokemon has something better. Why put Dig on your Krookodile when you can take advantage of Earthquake? Is Fly really necessary on Staraptor when Brave Bird does so much more? The goal of competitive is to win, and these moves don't often help further that goal.
Of course, there are some niche reasons you would want to use a two-turn move. Please note this isn't a holistic list and is intended to just give a few examples.
Some Pokemon, such as physical Dragapult and Brambleghast, don't have any other STAB moves and don't necessarily want to commit their Tera to get one. In this case, Phantom Force is the best move they can use, even if it sacrifices momentum. It certainly helps that Ghost is one of the best offensive types in the game currently.
Power Herb is a big reason to use charging moves. Power Herb + Geomancy is part of what makes Xerneas such a threat in Ubers, but it would be significantly worse without the Herb. In these cases, you sacrifice your item slot on the bet that the move will be enough to force progress in the game. Special Rock types like Nihilego often have to use this with Meteor Beam in order to have a reliable Rock STAB move.
Slaking's Truant Ability means it rests every other turn no matter what, so there's no added drawback for using a recharge move like Giga Impact. There's no extra recharge turn, so it can feel free to send off powerful STAB Giga Impact with few downsides (except that you have to use Slaking in the first place).
In Generation 1, recharge moves didn't recharge if they KOed the target. This means there's strong incentive to put Hyper Beam onto an attacker like Tauros, as it may give your team the wallbreaking power it needs to make progress.
In some niche metagames like 1v1, you just need to KO a single Pokemon. This makes the recharge moves much more appealing, as the recharge turn doesn't matter if you OHKO on turn one. Pokemon like Porygon-Z (Hyper Beam), Rhyperior (Rock Wrecker), and Primarina (Hydro Cannon), among others, can all make good use of these moves on some of their sets.
Two-turn moves aren't used often, and for good reason: they're inefficient, give up momentum, and are outclassed by popular moves like Liquidation, Brave Bird, and Earthquake. However, there are some times where using one might be the best play. But be careful -- losing the momentum could lose you the game.
