Mdthbs

Wild Shape is the prominent feature of the Druid class, but it allows only two uses until level 20, when Archdruid makes it limitless. This scaling feels quite odd to me and it appears to impact Moon Druids the most, since they rely on it for their combat prowess, while Land Druids have their superior spellcasting abilities.
I'm aware that it can be maintained for an amount of hours equal to half the druid's level and that the two uses are regained after a short or long rest, but the scaling of 2 uses between levels 1-19, then infinite at 20 feels weird and underwhelming, especially between levels 8 (when the feature gets the last power-up) and 20.

Is my concern sound? As a DM, I thought about increasing the uses by 1 at level 12 and 16 (total of 4). Would this make the Druid overpowered, outshining his party members (a Bard, a Barbarian and a Paladin) too much, either in fight or in utility contexts?

Note: the environment is pure core books.

Capstones (level 20 class features) are meant to be very powerful

Lets start with your first concern, while a 2-infinity chart would be silly looking for power progression the level 20 class features are all very strong and meant to showcase the player and character at the height of their abilities within the world. A level 20 druid can Wild Shape as much as they want without resting as both a sign of this power and as a nice reward to the player for making it to level 20. After level 20 there is no further progression to be had and you would expect a final, end campaign for the PC.

Are you using short rests correctly/often enough?

You seem to gloss over the importance that this feature recharges with short rests. Players should be taking short rests as often as they need/want to and you should generally just let them do it from a DM perspective unless there is a pressing narrative reason (ex: the invasion army is attacking at dawn) or persistent real threat. The party can and should be able to take multiple short rests during a day to catch their breath and bandage their wounds and recharge some of their class features to continue using them.

Wild Shape's change limitation is more about utility gating

Because a lot of the creatures a druid can wild shape into can have very specific helpful uses (a flying creature to scout, a small creature to sneak into places, a big one for a combat boost) the limit of 2 per short rest means that Druids need to think about how to use them most effectively for the situation the party is in. It also means that the party cannot solely rely on the druid for all of the possible utility of various beasts via Wild Shape so that the druid does not hog the utility spotlight.

It makes the druid a lot tankier, per combat

Giving the druid additional uses of Wild Shape effectively gives them extra hit points between each short rest, especially if they're Circle of the Moon and can use these extra Wild Shape activations as a bonus action.

Whether this makes them "overpowered" depends a lot on your campaign and encounter design (for example, it doesn't help with encounters where the party must achieve an objective in X rounds much, wherease it will help a lot in slugfest to-the-death combats). It will also let the druid take more risks.

What's the effect size?

At level 12, it gives them an extra Elephant (CR 4, which is 1/3 of their druid level) worth of health, or 76 HP (at AC 12, sure, but still 76 HP) at the cost of a bonus action.

For comparison, a cleric at 12th level has a single 6th level spell slot they could use to cast Heal (as an action, not a bonus action) for 70HP to a single creature.

This is a significant increase in how much damage they can take.

How about utility?

It also makes out-of-combat use of Wild Shape easier, since they can pop in and out of animal shapes (e.g. a spider to get through a keyhole, etc.) without running out of wild shape usages for combat as quickly.

Conclusion

It's probably too strong to just give them; however, if you really want to give the character something like this, an extra Wild Shape would be a reasonable Rare or Very Rare item enchantment; maybe something like a mask of the moon or +1 hide armor of the beast. You could also put restrictions on the extra wild shape slots, such as that it only lasts a round or two, or only lets them take CR 1 and less forms.