Can horses manage with three legs?

Not really. Dogs and cats can both survive fine in captivity with a missing leg - it's harder for cats.

Horses, because they are designed to be on their feet all the time, can't. A three legged horse requires a prosthetic to survive. Because this is extremely expensive, it's not always an option - and some horses can't recover. (Minis, because of their smaller size, often have a better chance). 3D printing has made equine prosthetics much easier to make and fit. Hind end prosthetics are easier than front end.

Horses with a prosthetic leg generally can't carry the weight of a rider or work. So, it's only generally done when the animal is a beloved pet. Mares suitable for breeding are also candidates - although they can't often carry the weight of the foal, embryo transfer is routine in horses.