Horses are not machines. You cannot just take the max speed of a horse and scale it up and go "that's how far a horse can go and how fast!" Horses are like us. The top speed of a human recorded is somewhere around 27mph. We sustain that for about 30-35 seconds, which is much less than a mile distance even at 27mph.
There is a guy who pulled 350 miles in something over 3 days running, but a. Humans actually have better endurance than horses (we're just a lot weaker) and importantly for this discussion, b. He was in absolute physical ruin at the other end - missing toenails, feed bleeding freely from multiple blisters, hypothermic, and woulda been in some Trouble if he hadn't had modern medicine right there to help him out. He was also running in the best equipment modern money can buy, on paved roads and clear terrain, and with his family driving alongside to help him swap socks, shoes, rinse the blood out, give him food to eat, water, and so on.
And again: humans are better at endurance than horses (honestly we're just fucking ridiculous, at peak condition, in terms of endurance, we're absurd) and the guy was a useless wreck at the end.
For horses, modern endurance races tend to max at 100miles, which are completed in around 18 hours (the max allowed is 24); there are some 2-day 100 mile races where you do two 50s and then add them together after sleeping overnight.
(This is the race for one rider with the same horse - there are much longer races (like the Mongolian Derby), but those involve switching horses, in order not to, you know, kill them.)
For the endurance races, those are intensely trained for and they're the big Effort for the year. Vet staff monitor the horses' very carefully at multiple stops, because it's very easy to injure them. Multi-day races are maxed at 50m/day in order to avoid injuring the horses.
After the race, horses normally need a month or more to recover from the effort, before it's even safe to do anything BUT rest with them, let alone actually race again.
Now if you're in a hurry and have no posts to get totally fresh horses (which is what things like the Persian messengers or the Pony Express did), you can eke a bit more out of this as within the story Théoden et al certainly did by having multiple horses, as it's harder for a horse to run with you on its back than to just run freely.
In the same time that it takes Gandalf to reach Minas Tirith, Théoden et al get to Dunharrow. Aragorn takes the Paths of the Dead because they cut under Dunharrow and vastly decrease the amount of distance in order to get to the Pelargirs, because his use of the Stone has shown him that if there isn't a lot more help than Théoden is bringing sooner than he is bringing it, Minas Tirith is fucked, and it is not physically possible for Théoden to get to the Pelennor before the 15th of March and still have horses that are capable of being ridden into battle.
(Obviously, part of what he did was cut off the enemy reinforcements by doing this, thus meaning that he reduced the need for significant more numbers arriving before the Corsairs would have . . . because he arrived instead of the Corsairs.)
. . . Shadowfax ran to Minas Tirith over the same terrain in four days. That would be a little over a hundred miles a night without stopping (Pippin is specific about how infrequently they stopped and that it seemed to have been entirely for Gandalf to talk to people) and when they get there he might as well have just come off a long rest, and thinks it's super stupid that he has to go wait in a stall while his two-legs goes and does talky-stuff. The next day he runs a super-speed race out to drive the Nazgûl off in order for Faramir to continue his retreat from Osgiliath (rather than be routed).
Shadowfax just did something that would kill a normal horse and then went " . . .what, like it's hard?"
To be fair, sure, Shadowfax was probably cantering because frankly fuckin' riding at a flat gallop for 4 days would be punitively exhausting even for Gandalf's new incarnation with some of his previous restrictions eased, never mind for Pippin, while a canter is p much the most comfortable gait. But for fuck's sake, guys.
Yes, a racehorse can, at flat fastest run, get over 25mph (Secretariat hit the 30s) - they do this for less than 2 miles.
A normal horse would have keeled over dead somewhere in the middle of that ride. If you had a super-trained endurance one they might make it in six-ish but they're not going to be useful for anything when you get there and are probably useless for several months if you want them to stay sound. (And you might still have killed them, or at least made them permanently lame.)
Shadowfax was like "toss me a waterbottle bro and then let's get back out there this stable shit is boring."
So yeah no, King of Horses was not, in fact, just kinda like the other horses.
