99% of the time, PCs abilities aren’t your concern, all you need to worry about is knowing if it sounds like they something they might be able to do. Worst case scenario just ask them to read the ability description.

Try these tricks (providing they are applicable!):

• Lower down the number of enemies under your control, but make them stronger. This way, you won’t have to deal with so many pawns.
• Don’t force your NPCs to fight to death. Assume that once, say 25% of your pawns are wiped out, their leader halts the fight and asks to parlay, or that if 75% of them/their leader go to the ground, the rest escapes.
• Tucker’s Kobolds.
• If a combat action leads to multiple dice rolls one after another, roll all of the dice at once - this speeds up the game.
• Use combat managers/helpers - Excel file, dedicated software, whatever helps you in math, use it.
• If the combat becomes too overwhelming, introduce some unsuspected elements. Sudden earthquake, volcano eruption, castle falling down, incoming of a truly powerful enemy, etc. Have some good in-game explanation for such an event!

Recommend you check out https://www.themonstersknow.com/

My tips:

1. let dice make decisions, and don’t try to explain them. Quickly number reasonable targets in a standard way (I do left to right, top to bottom), roll a die and move on. Get it down to less than five seconds.

2. it’s better to be quick than to be optimal. Get the attack out and move on.

3. enemy spell casters have three or four rounds to live. Mostly they fire their biggest blast in the first round or two, and then they’re trying to escape. Maybe they buff an ally or two, but that’s advanced strategy.

4. You handle a combat one step at a time. Work your way down the list, and each time its the same simple decision: which attack? which target? what result?

5. Sometimes combat in D&D is a grind. Go easy on yourself. Keep calling out the results, and the players will understand.

6. Six players is a lot. Delegate work to them, especially when they cause it (e.g. bane, or damage that ticks after their turn. ) it will keep them engaged.

What exactly is overwhelming about it? Your players should manage their own spells and abilities.

I got stuck on the same spot every single time I tried to make dnd work for my group. What finally worked for us was tokens. You dont have to go out and buy expensive miniatures - we use pennies, parts from other boardgames, coasters to represent area effects like guardian of faith, extra dice etc etc. The one other thing you need is a grid board. Someone in my group had one so we lucked out but I see them online for $20 to $30. Knowing where everyone is, how far you can move, who is gonna be hit by that cone of cold etc etc took off so much of the mental load and let us go back to having fun. You may already be doing this or it might not be right for your group but it helped us immensely.

The grid we use is dry erasable so we draw the battle map on there. We also use the markers on the pennies so we know the red marked one is the fire guy and so on. If you have any questions about this style of play lmk.

Lots of great suggestions but none really focussed on your concern about keeping track of magic in combat. If this is a question of keeping track of buffs, concentration, magical effects, or conditions, I found using “condition rings” on the players’, creatures’, and NPCs’ minis made this much easier. I started off with just the plastic rings off milk jugs or pop bottles, but fancy ones with the words written right on them make a nice gift request for a birthday etc.

Are you working from modules or from your own homebrew?

Since you have so many players maybe one of them can help you with these fights 💁🏻‍♀️