Do broken legs always mean a horse has to be put down or can the animal be saved? All the tv shows I've seen have portrayed that they can't survive a broken leg.

These days it depends on what the break is like. Anyone who paid attention to the plight of Barbaro the famous race horse will tell you he almost made it through breaking his leg in multiple places. Sometimes it depends on how far away the vet may be and if the leg can be stabilized.