rpg

3.5 huh? When rolling multiple attacks, do NOT roll them one at a time. If they get five attacks, roll 5d20s. Assign specific dice to the order of attacks if you need to or just go in ascending or descending order of the rolls; whichever is easiest for y'all. Similar with damage after you've determined which attacks hit. Just roll all the damage dice at once.

If your players are up for it, you can speed up combat by using an actual hourglass timer (30 or 60 seconds) to limit decision making time so they can't take 15 minutes planning their turn. This can also make combat more frantic and hazardous, of course, so you'd want to appropriately scale back some monster damage or HP if it becomes a problem and they put themselves in tactically terrible situations due to the pressure of the ticking clock. (This definitely needs buy in from your players first!)

As far as giving agency when it's not their turn, they can always shout suggestions or relevant tactical information to one another. You could theoretically let them help each other once per round or something, where maybe they roll a small die that gets added to the active players roll, signifying them shouting at an enemy to distract it or encouraging their teammate. This is basically 5e Bard territory, but there's no reason you can't use it generically.

I'm just gonna assume you're thinking of D&D, or something similar.

I think the standard thing to do when it's not your turn is plan your turn, so you can get straight to it when it is your turn. This also makes turns shorter, so you don't have as much time to fill when it's not their turn.

If you wanna build the habit, start getting into the swing of "Alice, it's your turn. Bob, you go next, so get ready. Alice?"

Assuming we're talking about D&D 5e... this is a common issue.

More than 5 or 6 players will bog down combat exponentially.

Martial character options like Action Surge, Extra Attacks, and weaponized Bonus Actions will of course extend turn time. You can speed things up by telling the player the target AC up-front instead of the common guessing game: "does 14 hit? ok I'll use Inspiration to add d6, does 16 hit? Ok, next attack, does 18 hit...?"

Players being tuned out, needing a battle conditions summary in order to decide what to do each time their turn comes up... bad manners but are you really ready to tell an otherwise beloved friend to shape up or don't come back?

There's a fairly common condition, aphantasia, which is basically inability to imagine visually. Using a miniatures battlefield instead of theater-of-mind for more complex battles can help players with aphantasia stay on top of battle conditions.

Players who perpetually pitch outside-the-box, game-breaking interpretations of their abilities, like trying to use Acrobatics proficiency like a free Spider Climb spell... yep that taxes time even if the ruling is "no that ability doesn't work that way". Putting a time limit on each player's turn might help quash time-wasting spitballing.

Sometimes I get players to make a series of D20 rolls for me and note down the results. Then I use those for their opponents in a later round. Not in the order I get them - usually every second or every third.

What game are you playing?

Are you locked into that system?

Make sure your players are planning their turns in advance. That way they have something to do when it's not their turn, and turns go much more smoothly so there's less time in between them.

Reduce your mental load to increase your speed: limit yourself to fewer additional stat blocks per encounter (e.g. if there is a boss battle with lair actions, you should ideally have 1, at most 2 additional monster stat blocks on top of that boss). Running something like 4 stat blocks is sufficient cognitive load that there is no way for you to play your monsters optimally and still be fast.

Reduce your reliance on small dice rolls to increase your speed: preroll a table of damage for any given statblock's actions, so that you can dole it out quickly (still roll the to-hits live tho, in case of crits and misses and reactions, and have misses consume a line from the preroll table too).

Add a low-value off-turn minigame for players to play: I like to make knowledge checks free (History, Nature, Insight, etc). You could allow players the chance to each get 1 free knowledge check in per round, but only off turn. That way if it's another player's turn, and they are thinking for a while, you can have the other players roll their knowledge checks and tell you what they are looking for. Resolve them later, at the beginning of those PCs' turns, which will also keep other players engaged since they are learning new info. Don't let them ask tho, save it for when you want to provide the distraction.