Honestly, I think people often over-think handling their new dragons a lot. i've seen everything from people literally not touching their new dragons for months (which is totally counterproductive and after too much time goes by you miss your window of opportunity), to them actually doing the "Putting their hand inside the tank and just holding it there for 20 minutes" thing, which is actually a hand-taming technique to train pet birds/parrots that were not hand-raised but rather parent-raised and have no concept of human hands...
Bearded Dragons are often mistakenly called "Beginner Reptiles" or "Good Reptiles for young children", and these label certainly did not come to be because Dragon's husbandry is easy, lol. These labels came to be placed on Bearded Dragons because of their docile nature and their wanting/needing, or at the very least tolerating interaction with people.
I grew up in a bird-breeding family, and I myself bred several species of parrots for about 20 years. With birds it's quite simple: If they are pulled from the nestbox by the age of 2-3 weeks old, totally separated from their mothers and/or fathers, and totally hand-fed and hand-raised by a person or people from that age forward, then they are pretty much going to be fine with people for their entire lives, barring any neglect or abuse. If a baby bird is not hand-fed/hand-raised, or even if they are pulled from the nextbox too late (say later than 4 weeks at the very oldest, 3 weeks is really the accepted cutoff), then they are not going to innately trust people and you're going to have to really work with them, every single day, until you get them to trust and accept you, and often times these birds cannot ever be tamed (take for example the American Budgies/Parakeets you can buy at Petco and PetSmart, they are raised by their parent birds in large warehouses in groups of thousands, and they never even see a human being until they are about 8 weeks old, when they are pulled away and put in a cardboard box and shipped to the pet stores, so imagine how difficult it is to tame them)...
In contrast, Bearded Dragons that come from mass breeding vendors (all Bearded Dragons that are sold at any Petco or Petsmart come from mass-breeding vendors) and that are not ever handled by people or have any interaction with people before being shipped to the pet shop, well the 2 that I've brought home from Petco at about a month or so old were sleeping on my chest the very first night in my house...Now that's not anything I did, nor is it anything that anyone did, as I was told by the Animal Manager at my local Petco, their current Northeast vendor ships all of their babies as soon as they reach 4" long or 4 weeks old, whichever comes first, which accounts for how tiny they are at Petco. And these babies have never been handled, touched, talked to, etc. until the day they are packaged up. But in-contrast to birds, Bearded Dragons just naturally have a docile nature.
Now you have brought home a Dragon who is 3 or 4 months old, so this could create a bit of skittishness, especially if he was mishandled by anyone prior to you bringing him home...That being said, yes, he's going to be jumpy and flighty, trying to jump out of your hand and such, but the easiest way to handle this situation is for you to JUST SCOOP HIM UP FROM UNDERNEATH!
Seriously, that's what you need to do. Usually in this situation, at least half the issue is the nervousness of the new owner. You need to talk to him softly as you approach his tank, tell him what you're doing, keep talking to him softly as you put your hand in the tank, and the entire time while you show him your hand, so he associates your hand with your voice and he knows that the hand belongs to you, and then after he sees your hand, JUST SCOOP HIM UP AND PUT HIM AGAINST YOUR CHEST AND HOLD HIM AGAINST YOUR CHEST WITH YOUR HANDS. Then sit down on the floor, carpeting is best, that way if he jumps out of your hands he won't have far to fall. Make sure you do this in a room with a closed door and nothing that he can get underneath, you may have to block off an area around you on the floor in a secure room. Then just literally scoop him up and start handling him.
He's most likely not going to bite you, I've only been bitten once by a Dragon, and that was an adult who I was syringe-feeding Critical Care at the Rescue I work at, and it was actually my fault for getting my hand too close to his chompers, lol. And this was a big, male, adult Dragon who weighed over 500 grams. It didn't feel good but it didn't really hurt, it did break the skin, barely, I didn't bleed or anything, and that was it. So your juvenile Dragon is most likely not going to bite you in the first place, he's just going to be a bit jumpy. Use both hands to keep hold of him, and once he settles down then try stroking the top of his head, neck, back, etc. (in the direction of his scales, no against), under his beard (some like this, some don't), the entire time talking to him. You obviously need to do this every day, several times a day, but that's what you have to do, just go for it.
Sitting and waiting for weeks and not touching/handling your new Dragon is not the best way to get him "used to you", some people won't agree with me, but again, I've owned 5 Dragons total, 3 were my own that I purchased as 1-2 month old babies, the other 2 were sub-adult, male rescue dragons who were sick and neglected (the one I just took in 3 weeks ago today is a stubborn little guy, but we're getting there, lol). And all 5 of them I just picked up from the first second and there ya go.
So the answer to your question as to "How are people who have skittish Dragons giving them baths if they can't even handle them?" is simply that THEY ARE HANDLING THEM, LOL. Just because they're "skittish" does not mean you shouldn't handle them and interact with them. In fact, it means just the opposite in my opinion, the more skittish they are, the sooner and more often you need to handle them...again, letting a "skittish" dragon go for days or weeks without being handled is counterproductive, and probably at least half due to your own nervousness, not the dragon's...So just go for it!