There's a difference between slow and draggy. I find 5e combat to be horribly draggy while not being that slow by WotC D&D standards, The difference is that slow is the length of time a combat takes as measured by the clock, draggy is about how interesting and tense the combat is in that time. It's possible to have a two hour combat that doesn't drag because everyone is on the edge of their seats and to have a half hour combat that does because it's just so tedious.
5e combat is IME both faster and draggier than both 3.X and 4e combat. This is for two reasons:
- "Bullet-sponge" enemy design
- Tactical position being made almost meaningless deemphasising tactics
Other people have started to mention the bullet sponge design. Thanks to bounded accuracy enemies have a narrow range of AC and saving throws, meaning that the biggest way most NPCs have to protect themselves is their hit points. You'll normally hit them and have to chew through their hp until they hit zero. In AD&D your average ogre had 19 hp. In 3.5 your average ogre had 29hp - but you were much more likely to have a bonus to damage from your stat. In 5e? Your average ogre has 59hp and an AC of only 11. Even a first level PC is probably hitting that on 6s - the goal is just to chew through the enemy's hit points as they stand there. Bullet sponges are known to be one of the least engaging ways of increasing difficulty in a video game (having to pump more shots into the enemy) and it applies in RPGs too.
Meanwhile tactical positioning barely matters in 5e. In 3.5 and 4e flanking is a thing - move to opposite sides of the enemy (a risky place to be) for +2 to hit. It encourages you to try to move into certain spots and out of others. 5e drops flanking to an optional rule. In 3.5 and 4e you get significant bonuses by trapping the archers and the casters in combat; in 3.5 they are forced to switch from using Dex to Str to hit, and in 4e artillery NPCS get +25% baseline damage at range and -25% baseline damage in melee. In 5e, thanks to the weapon finesse rules being a default the NPC archer who draws their shortsword for melee is using exactly the same stats and just drops their damage die from d8 for the longbow to d6 for the shortsword while still using dex. W00t. And forcing the caster into melee? Just as pointless - they take disadvantage on spells that make attack rolls, so they instead use spells that force saving throws at full effectiveness. The only real part of tactical positioning that reliably matters is focus fire.
TSR-era D&D was based on a paradigm of combats being fast to resolve. There wasn't much you could do in them by the rules except pick spells, but when a fight happened it was intended to be fast. In the game of "see whose health bar drops first" a whiff is easy to resolve and when the health bars dropped they dropped fast. 3.X and 4e D&D were based on a paradigm of combat being engaging and you had time and options to counter-play the enemies and rescue each other. 5e is somewhere between the two a level of tactical engagement that's about 80% of the way towards TSR era D&D and a level of speed that's about 75% of the way towards 3.X and 4e. I find this an unhappy medium.