Our DM started to master for us about two years ago, using published materials. We played Lost Mine of Phandelver, then Waterdeep: Dragon Heist and now Dungeon of the Mad Mage. All of this is good.
Now, the problem: he needs a lot of time to resolve what is happening. There are multiple factors contributing to this, I'll focus on two of them
Desire to play by the book. He tries to keep to rules as written and within that optimize. He is counting squares of movement that happens off-screen in combat situations, takes a lot of time to measure exact distances and movement. Using a battlemap on Roll20 has made any combat a tactical mini-game. (To be fair, the players also protest if he gives the monsters more leeway on things like movement, while watching ours like a hawk, certainly pushing him to try and adhere to the rules).
Desire to optimize tactics. As a player he is a min-maxer, and as a DM he approaches running the monsters the same way. He tries to optimally use their abilities and the environment for tactics to make our life challenging. looking up monster abilities and "The Monsters Know What They are Doing". This is especially the case if the monsters have many abilities or spells at their disposal. (He also has a hard time keeping apart what HE knows vs what the monsters know, typically he tries to have them take the objectively most effective action, which multiplies the factors he has to keep in mind).
Between all these, it is quite common for us to sit around for minutes on end while he is leafing through stuff.
We mostly play online on Roll20, and being just a single click away from all the other seductions of entertainment to bridge the boring lulls in the action is testing at least my self-discipline every session. One of the other players, who has a physically taxing day job, has fallen asleep during play repeatedly.
I used to DM for our group for years before he agreed to DM, as I really wanted to play as a player. He puts a lot of effort into preparing, and he is a close friend, so I want any help I can extend to him to be as encouraging and positive as possible and I don't want to patronize him.
I've talked with him about how it is challenging for us to wait for things to continue, but this unfortunately has not helped him solve this. So instead of just bringing this up again, I want something to suggest that we could try out.
I think he needs concrete things he can do to help him. I find it easy to improvise as a DM to keep things going, while he aims to stick to the written adventure, so I don't have a lot of useful practical help for him on what he can do, other than reading the module beforehand and marking it up.
Can anyone help with concrete DM procedures, methods or tricks they use to be able to keep the game flowing, while sticking closely to published material? (We are playing D&D 5e, but I think answered do not need to limit themselves to 5e).
How can he run combat with monsters faster and more efficiently without spitballing when it comes to rules adherence?