No, they're not. One could argue that some of the recent breeds, designed for a particular aesthetic rather than practicality, are, but that's not really within this subject.
> horses are big and react poorly to being wounded: a solid arrow hit on a horse is very likely to disable both horse and rider. And while light or archer cavalry might limit exposure to mass arrow fire by attacking in looser formation, as we’ve discussed, European heavy horse generally engages in very tight lines of armored men and horses in order to maximize the fear and power of their impact. Unsurprisingly then, we see from antiquity forward, efforts to armor or protect horses, called ‘barding’: defenses of thick textile, scale, lamellar, and even plate are known in various periods, though of course the more armor placed on the horse, the larger and stronger it needs to be and the slower it moves. Nevertheless, the size and shape of a horse makes it harder to armor than a human and you simply cannot achieve a level of protection for a horse that is going to match a heavy infantryman on the ground, especially if the latter has a large shield.
Curious, then, that the golden horde was stopped by a conflict regarding succession and not heavy infantry. Sometimes it was also slowed by fortifications, but learned quite efficient siege tactics and how to build siege machinery by conquering parts of China and the nearer East.
European medieval cavalry warfare saw the development of sophisticated horse training and group tactics, which mitigates the risk that a horse would bolt at the first wound. The reason infantry with poles were the default counter against cavalry, and not bowmen, is that these weapons wound more deeply and stand a chance to overcome horse training or bleeding out the horses.
If you could flank cavalry with bows it was still a good idea to do so, largely because cavalry knights were heavily armoured and in formation, i.e. not very agile and typically locked into shock advancement. It's important to keep in mind that medieval european destrier horses were big but not tall, unlike the breeds popular among contemporary european militaries, which tend to be slim and tall.
Until mechanised warfare became dominant european warfare was highly bound by tradition and ceremony and honour, rather than efficiency. This is an important reason for the quick fascist advancements in early WWII, and the success of highly mobile mongol cavalry. Previously european militaries typically decided on whether to meet on a field and clash there, or whether one party were to retreat into a fortification and turn it into attrition warfare. Skipping past the enemy Schwerpunkt with a mobile force was more or less frowned upon, and you can see this even in modern military thinkers like Clausewitz or turn-of-the-millenium US warfare.
Deciding on where the field is, where the enemy officers are concentrated as well as their troops and entering into an honourable, decisive battle there has been the dominant mode of european and european-descendant warfare for a long, long time. It hinges on soldiers (and horses) to be trained to not bail out in the face of danger or light injury, and wouldn't be possible at all if horse mounts were frail.
No, they're not. One could argue that some of the recent breeds, designed for a particular aesthetic rather than practicality, are, but that's not really within this subject.