Learn Turkish: Sağlık için diyet yapıyor. - She is on a diet for her health.

In Turkish, many activities use the helper verb yapmak (to do/make). While in English you "go on" or "are on" a diet, in Turkish you literally "do a diet" (diyet yapmak).

The translation says "for her health", but I don't see a possessive suffix on sağlık. Why is that?

In English, we naturally say "for her health". Turkish often leaves out possessive endings when speaking generally or when the context is obvious. Sağlık için literally means "for health." If you specifically wanted to emphasize her health, you could say sağlığı için.

How does the word sağlık break down?

The suffix -lık turns adjectives or nouns into abstract nouns. Here, it attaches to sağ, which means "alive" or "healthy" (as well as "right"), to create sağlık (health).

Does diyet mean diet as in "everything I eat" or just "a restricted diet"?

In Turkish, diyet usually means a restricted eating plan to lose weight or manage a medical condition. To talk about general everyday nutrition, you would use the word beslenme.