BTW, here is a neat set of random tables for Conjure Animals:
https://docs.google.com/document/d/1TAUqgYfi4gtDC6WOH41Y2JW-Wjh6NOx-Oq-D92pM4EM/
It gives a different random table for each CR type and terrain.
You zoomed right past, then quoted exactly the problem I am having.

It is strong. Powerful in the way combats are a load of no fun.
Sit back while 8 giant badgers take on a handful of orcs, or 8 giant owls swarm all the bandit archers, or 8 elks charge into the fray ... basically all of the eight quantity of 1/4 CR creatures joining the party turns it into a rout. It's been a powerful game changer that sucks out fun. We have gotten the idea that 3rd level spells are a new level of power, but this has been a bit much to the point the party is getting the idea the druid could finish off most encounters herself without help from the rest... for now. Shr
But I think one of the early responses reminds me of the most useful, and in our case quickly forgotten by us 5e novices, Concentration. I'm almost certain we're forgetting to make the caster deal with the damage she's taking while she's conjured the beasts.
The other thing to be aware of is that 5E has kind of a rock-paper-scissors situation going on with monster design. Swarms of monsters or PCs are hard to deal with on a single-target basis, but that's exactly where AoE (area of effect) shines. A CR 1/2 Magma Mephit's 2d6 breath weapon looks weak at first, but if it hits 5 targets on a strafing run, and then the Magma Mephit flies away to possibly do it again in a few rounds, it looks quite a bit better, especially if there are five other Magma Mephits doing the same thing.
Generally speaking, if you want to build an adventure that can challenge swarms of henchmen/conjured animals, you need one or more of:
(1) Environmental effects that inflict damage proportional to the number of creatures. Ravines that must be crossed on a narrow path (e.g. Acrobatics DC 12 to avoid 5d6 falling damage), foul air that inflicts 1d6 poison damage per ten minutes, etc.
(2) Creatures with AoE weapons.
(3) Highly-mobile creatures that choose when/where to engage. Dragons are especially good because they are #2 and #3 at the same time, plus they are generally quite stealthy. A Young White Dragon can quite quickly deplete a whole flock of Giant Owls with its breath weapon using strafing tactics, if the owls simply attack it in the air.
Downside: if the Giant Owls are being used as flying meat shields to delay the dragon while PCs deal damage with ranged weapons (especially Sharpshooter fighters/ranged), then the PCs will still come out ahead of the exchange with the Young White Dragons. Conjure Animals is indeed a very strong spell, especially if you manage to get flying animals like Giant Owls to appear. (Some DMs use random tables based on terrain, others simply honor player requests.)
Upside to the downside: the PCs are
supposed to win. All you need to do as a DM is provide sufficient challenge to (1) be fun, and (2) explain why someone else hasn't already killed the dragon using similar tactics previously. If players effectively use combined arms and giant owl swarms and trained Sharpshooters to trap a dragon in a space from which it cannot flee before getting torn apart, I'd say the player earned their victory. 5E isn't an adversarial game of DM-vs-players after all--it isn't chess.