Something I never cared for in Dungeons & Dragons was the way initiative seemed designed to slow down combat for no good reason. The idea is that every combatant (or at least every player character), rolls to see where in the order of combat his or her action comes each round of combat.
In the old days, this is how Gygax managed bigger, slower weapons that did more damage versus smaller, faster weapons. Upon looking into D&D 5th Edition, I was surprised to find initiative still used at all. By 2nd Edition, everyone I know had abandoned it.
Some people make or buy fancy initiative trackers that we would have laughed at back in the day.
A better system is side initiative, but some people seem to think this means all the combatants on one side act and then everyone on the other side acts. But that’s not a good system.
The better way to manage it is this:
At the start of combat, each side rolls 1d20 and adds the number of combatants on their side.
If the monsters win, the DM decides which one acts first and deals with that. Then, if it attacked, deal with the player character it attacked. Then proceed clockwise around the room: monster first, then PC.
If the PCs win the initiative, they decide which of them is ready to act first, and the DM deals with that and then the monster that character attacked. Then play proceeds around the room, PC first, then monster.
The next round just continues in this way, without rolling for initiative again.
If a spellcaster casts an area-effect spell or if there is some other disruption, the DM should consider if it causes the combatants to pause (especially if some have to make a saving throw). If so, then both sides should roll initiative again.
Deal with characters who have no opponent or extra attacks at the end of the round or in the middle, as makes sense.
The idea here is that the more people you have on your side, the more likely it is that someone will act quickly. The speed of your weapon doesn’t matter when considering when you act; that instead affects how likely you are to hit, with small weapons getting a +1 in my system because they are quick enough to get multiple strikes. Remember that “an attack” isn’t a single strike but a series of feints, parries, shield buffs, and strikes.