Its all about technique.
My normal process is :
- set gear in highest (11T cog), open rim brake caliper, undo QR, and remove wheel from bike
- remove cap and lockring from presta valve, put aside.
- quick visual inspection of tread looking for cause of puncture
- use two plastic tyre levers to remove one side of tyre bead from rim
- Pull tube from tyre, inspect.
- finger inside of tyre looking for a sharp, and remove anything found.
Fitting new tube:
- fit new valve stem into rim, add lockring loosely so it doesn't fall back inside rim while working. Lay rim flat like a pizza.
- Tuck the tube into the tyre evenly all around
- Then start at the valve and tuck the tyre's bead into the rim. Work around the rim till it gets tight. At this point some people put a small puff of air into the tube, but that's never worked for me.
- I use fingers on one hand to shove the tube into the tyre, and then I use the other hand to squash the two sidewalls together to pinch the tube lightly so it can't be in the bead area. Then I push an inch or two of bead over the rim, and repeat.
- If I see tube stuck between rim and tyre then I back off and reset. There's nothing to be gained by trying to recover.
- When tyre is fully in the rim, I push the last segment over and visually look between bead and rim valley wall - a save here means the tube is still okay but if you'd started inflating the tube can be torn.
Note - some tyres are just tight, and some rims are oversized by the tiniest amount. Combinations like this can be super hard to fit.
If you're putting a too wide tube in, that can be difficult. Likewise a tube that is too long may need to be redistributed to fit in - you don't want folds anywhere.
Some people use soap or water or similar as a lube, but I find they just make the job messier.