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.