Turn Structure
A game turn should be simple, easy, and interesting. It should not be a list of steps and procedures.
After complexity, this is probably the #2 criticism I have of other designers' games I playtest.
This is a huge mistake I see in game expansions. Have you ever felt like a game was simpler and better without its expansion? That extra step each turn/round has a cost.
No phases
A turn should not be composed of "phases" unless absolutely necessary.
Make a turn about a single, consequential action. Remove and minimise everything else.
Turn structure is a very high design priority. Redesign other parts of your game, in order to simplify it. Preferably, begin your design with a simple turn structure, and build the game within it.
Simpler turns are easier and simpler, but they're also quicker. In my experience, boring games are usually boring because of downtime, i.e. waiting for other players' lengthy and complex turns.
A poor game designer argues the case as to why these steps are necessary. A good designer abolishes them, and then finds other ways to make the game work.
For simple games, this is even more important. You cannot have phases in a wide-appeal game.
There's a simple test here, for you and your playtesters: are there parts of the turn that you keep forgetting, don't actually care about, or are a drag? If so, your turn structure is wrong.
Limit choice
If a player has to make a choice, they'll take time to think about that choice. They'll do this, even if the choice isn't important.
Steps in a turn are bad. Steps with choices in them are far worse.
Try to confine the choices to one part of the turn, and make those choices important. Make the rest of the turn automatic and choiceless.
In my gangster game, drawing one card per turn gave too many cards, so I let players buy one for a dollar, at the end of each turn, if they wanted to. This step then contained a decision, which slowed the game right down. I eventually decided that the players must simply draw one card, with no choices. (This was me working backwards from this outcome, not justifying why it had to be as it was.) One card per turn, however, was too many. The hand size and complexity was excessive. To reduce the complexity, I made half the cards simple "Package" cards. All you could do with these was collect them, and trade them in in bulk, for resources. These cards were super simple, identical, and therefore didn't add complexity to the players' hands. Also, the annoying and slow choice step was gone.
Consider the difficulty and speed of a step. A simple and automatic drawing of one card per turn is usually worthwhile.
The Magic card "Sensei's Divining Top" lets you look at the top three cards of your deck, and rearrange them. Players would use it once or more per turn. Its benefit was small, but required extensive thought. The card was eventually banned.
If you must have choices, then go into the choices themselves, and minimise them.
In my farming game, I had ten little abilities. Instead of making a space that let you choose three of them, I just bundled them into groups of three or four. That was enough choice.
In my pirate game, you'd roll the blue die each turn, and get a beneficial effect from it. I wanted to include a "Merchant" effect, that let you sell cubes. I didn't say "you may sell cubes for 2 each", I just said "you may sell 2 cubes for 5". This was a simpler and faster choice.
Cut out tiny tasks and choices from players' turns.
In Ticket to Ride, there are five train cards, and on your turn, you can take two of them. However, once you take your first card, you draw a new card to replace it, and then you decide which second card to take. This means that you have to consider if you want the new card. This adds a second small decision and a task to the turn. I would've just had the player take two cards, THEN replace them.
Get rid of maintenance
I just want to play the game!
A player's turn should just be about doing the thing (or things) they want to do, plus minimal (ideally zero) maintenance and bookkeeping.
If there are tasks that must be done, to maintain the game state, try extremely hard to move them from the turn to the round.
Having to put out ten tokens at the start of each round is not great. Having ten turns, in which the player must also put out one token, is far worse.
Even better, get away from having "rounds" entirely. Rounds tend to exist because a game needs to undergo housekeeping. Does your game really need that housekeeping?
If you need maintenance, try to blend it into the actions themselves.
In King of Tokyo, there are always three special power cards on sale. Maybe no one wants those cards. Instead of having the cards be discarded and redrawn regularly, a player can simply pay two energy during their turn, to reset the cards.
In one of my prototypes, players moved to spaces around a town, that had various effects. To stop a player using a spot repeatedly, and encourage them to move around the board, the player put a "depletion" counter on a space when they used it, so it couldn't be used again. I wanted to get rid of this step.
I removed the depletion counters, and replaced them with "stash" counters. These added goodies to certain spaces on the board, to encourage players to move across the board, to get them. I needed far less of them, as just a few stashes would keep the players moving around all game.
In the same prototype, the player would also draw a "Change" card each turn. These cards moved the non-player characters, advanced the time, removed those depletion counters, and made events happen. I moved all these cards into other decks. You'd occasionally draw one, do some maintenance, and then draw a new card.
Player-controlled gameplay
Try to keep game-controlled effects out of the game. Yes, you can have "event cards" that happen periodically, or "neutral players" that perform actions, but does the game need them? If they're needed, make them very simple and quick. Far better to let a player control the "bad guy" momentarily, and use it to attack their opponents, than to have a whole process for deciding what the bad guy does.
In one of my prototypes, I decided that you could get a bad "reputation". But if you had a bad reputation, what did that mean? I could've had neutral characters who would periodically chase you, or detect you and react to you. I decided not to do that. Instead, it's the other players who can target you, and steal your points, if you have a bad reputation. It's far simpler, and requires no maintenance.