Welcome to wirenuts.
My first six splices did that too, and I undid them and redid them until they were right. Since then, I've done thousands that are solid. It's a learning curve.
Use the right size wirenut. For three #14 or #12, yellow or red. Each wire-nut has a spec sheet, with an allowable range that's surprisingly broad. Work in the center of the allowable range. Near the edges, they are too finicky.
Use newer wirenuts. Reusing them isn't the issue. It's that technology marches on, and they've been made vastly better, so a 20 year old new-old-stock wirenut pales by comparison to a scavenged 2-year-old one.
Sometimes a particular wire-nut just won't perform. Toss fussy or damaged ones. Sometimes they have a scrap of wire stuck in the bottom; those will never work.
Don't throw defective wirenuts on the floor, where they'll be confused with dropped good ones. Put them in your pocket then to the trash.
Use quality wirenuts. As in all things, there's a lot of cheap dreck.
Clean up wires, if they are too-bent or have loose strands going all over the place. No need to pre-twist solid wire, but pre-twist stranded wires in the same direction as the nut spins.
Don't aim for this, but if a bare wire pokes out a bit when you start, don't worry about it. If it pokes out when you're finished, undo, trim and repeat.
Square up the ends of the wires, so they aren't going in staggered. No need to pre-twist. Proper use of a decent wire-nut will do the twisting for you. Pre-twisting in the wrong direction is worse than doing nothing.
If you're struggling with them, pre-tape the 3-4 wires together an inch back, on the insulation.
Turn the nuts the right direction.
Crank them down earnestly. Normally I say "monkey tight not gorilla tight", but they are small and hard to grab, so I say "just give it the works".
Yank each wire hard, separately. If a wire comes out, see above.
Don't use tape to hold wires in. If the wires won't hold, do it again, right. Use tape to keep the thing from unscrewing (unlikely if done right) or to keep anything from poking under the wirenut and contacting a conductor. Remember when finished, insulation should go under the sleeve a bit. Don't go crazy with the tape. I don't use it. A connection that fails without tape has already failed.
Think twice for connections larger than 10 AWG. It's too hard to crank them down hard enough. Better is with real terminal blocks, and torque to spec. And remember the goop if you're using the new AA-8000 aluminum wire, which you should consider for anything over 6AWG.
Speaking of that, never wire-nut aluminum. Learn what to do with aluminum wire.