Sped up Pokemon are not given multiple turns in a row - in actuality, they're allowed to act during extra rounds. One "round" is the leader's turn, then the turns of your partners, then the turns of your enemies. After the round for Pokemon with normal speed is finished, a round for Pokemon with quadruple speed starts - during this round, every Pokemon's turn is skipped unless it has quadruple speed. After this, there's a round designated for Pokemon with double speed, then there's one for triple speed, and then the round for Pokemon with normal speed begins.
Example if you have two partners and your first partner is at triple speed:
Normal Speed Round: You move, partner A moves, partner B moves, enemy moves.
Quadruple Speed Round: nobody moves.
Double Speed Round: Partner A moves.
Triple Speed Round: Partner A moves.
Repeat.
So if only your leader is sped up, it'll appear to have multiple uninterrupted turns in a row, but if a partner or enemy is sped up, its extra turns will be interrupted - and, if multiple Pokemon are sped up, they will interrupt each other's turns, so if everyone is at quadruple speed, the game will play as if everyone is at normal speed. It's weird, but it makes sense now, doesn't it? Great, here's three other rules which overcomplicate everything: the swapping places penalty, immediate speed buff turns, and the action penalty.
First, the swapping places penalty: when you swap places with a Pokemon, its next turn is skipped. When a pokemon swaps places with a pokemon who acts later in the round, the order makes sense: leader moves, partner moves, enemy moves, leader swaps with partner, enemy moves, leader moves, partner moves, enemy moves. But when a pokemon swaps places with a pokemon who acts before it in the round, it looks weird: leader moves, partner swaps with leader, enemy moves, partner moves, enemy moves, leader moves, partner moves, enemy moves. The second example is never encountered in the unmodded game, just in my hack, since normally the leader is the only Pokemon which is capable of swapping places. It makes sense when you think about the fact that the Pokemon you're swapping with is also taking its turn at the same time you are, as it's moving to the tile you used to be on, and that cost needs to be paid.
Something else to note is, when you give/take an item to/from a partner, its next turn will be skipped. You won't get your next turn skipped if you give/take an item to/from yourself.
Now, initial turns - this is how it works in the original EoS and RRT, without my hack. When you eat a quick seed or use agility, you'll get three "immediate turns" even if you're just going to double speed, but if an ally uses agility, it will only get one. This is because, on the turn a pokemon gets a speed buff, its turn is immediately repeated. And for the leader only, something else happens - when you get a speed buff, the current round becomes a round designated for Pokemon with quadruple speed. Here's what it looks like:
Normal round: you eat a quick seed. The round is now a quadruple round. You get your turn repeated (first extra turn). Partner and enemy turns are skipped because they don't have quadruple speed.
Double round: you move (second extra turn).
Triple round: nobody moves.
Normal round: you move (third extra turn), partner moves, enemy moves.
Quadruple round: nobody moves.
Double round: you move.
Triple round: nobody moves.
Repeat from the last normal round.
In my hack, I removed the part of code which sets the current round to a quadruple round, because it bugged the game up during manual mode. How my code works is, after the leader's turn, my code changes the leader to your next Pokemon, then jumps back in to the leader's turn again. After playing the leader's turn for every one of your Pokemon, it skips the part of code which does the turns of your partners, so they don't get extra turns where they're controlled by AI. This means your partners have a few of the rules which are exclusive to your leader during manual mode - for example, before I fixed it, if your partner died during its turn from a self destruct move or poison while you were controlling it, you'd get a game over. The rule which set the current round to be a quadruple speed round screwed with the turn order a ton - before it was removed, it looked like this:
normal round: you move, partner A eats a quick seed, partner A gets its turn repeated, round becomes a quadruple round so partner B and enemy turns are skipped.
Quad round: nobody moves.
Double round: partner A moves.
Triple round: nobody moves.
normal round: you move, partner A moves, partner B moves, enemies move.
Effectively, when a partner ate a quick seed, every Pokemon which moved before it in the turn order got an extra turn. I always thought getting 3 extra turns for eating a quick seed was cheap in the original game, so I didn't feel any remorse in removing it.
Finally, there's the action penalty. When you use an action other than moving to a tile, all of your next extra turns are cancelled. So if you have quadruple speed, turn order looks like this:
Quadruple round: you use tackle. You used a move, so your extra turns are skipped.
Trip/double rounds: nobody moves.
Normal round: you move (if you attack here, it will cancel your extra turns), your partners move, your enemies move.
Repeat from the quadruple round.
So, no matter what speed level you are, as long as you have more than normal speed, you only get to attack twice before enemies move, IF you make sure you attack during an extra turn - otherwise, you get no extra turns if you're constantly attacking. This means, if you want to attack twice in a row, you need to... Move to a tile or wait on your normal speed round. This effectively makes most of this information, and speeding up itself, pointless, save for niche situations. Damn.
Also, don't trust the "You returned to normal speed" messages. The counter which keeps track of how many turns until your speed lowers a level is decreased at the start of a turn. Because of this, your speed can return to normal at the start of your double speed turn: so if your leader is running out of sped up turns, the game will tell you "you returned to normal speed", when you still get to have two turns in a row. It's extremely confusing, especially if you're in a life-or-death situation where you need to know how many tiles you can move, but if you pay attention and keep count, you can use this information to its fullest and get out of seemingly impossible situations. Or stall your death by a few turns.