Learn Hardcore Turkish: Ben staj yapıyorum.

You can drop Ben. The verb ending already shows the subject, so Staj yapıyorum is a complete, natural sentence. Include Ben for emphasis/contrast or when you want to be extra clear (e.g., Ben staj yapıyorum, o çalışıyor.).

Where is the “am” in “I am doing”? Why isn’t there a separate “to be” word?

It’s built into the verb ending. In yapıyorum, the ending -um marks the 1st person singular (“I”). Turkish doesn’t use a separate “to be” with verbs; person and tense are expressed with suffixes.

Why is it staj yapıyorum instead of a single verb like “to intern”?

Turkish often uses a light-verb construction with yapmak (“to do/make”) to express activities: staj yapmak = “to do an internship.” Similar patterns include alışveriş yapmak (to shop) and spor yapmak (to exercise). There isn’t a commonly used simple verb meaning “to intern.”

Can I say staj etmek?

It’s not idiomatic in everyday Turkish. The natural choice is staj yapmak. You may also see staj görmek in some contexts (especially in education/medicine), but staj yapmak is the safest, most common option.