Learn Swedish: Jag insåg att jag hade gjort ett misstag. - I realized that I had made a mistake.

We use the past perfect tense to show that one action happened before another action in the past. First, the mistake was made (hade gjort), and then later, the person realized it (insåg). You form this exactly like in English: with hade (had) plus the supinum form of the verb, which is gjort.

The verb insåg looks a lot like såg (saw). Are they related?

Yes! Insåg is the past tense of inse (to realize). It is a compound word built from in + se (to see). Because the past tense of se is the irregular verb såg, the past tense of inse becomes insåg.

I learned that göra means 'to do'. Why is it translated as 'made' a mistake here?

While English has two different verbs for 'do' and 'make', Swedish mostly just uses göra for both. When you make a mistake in Swedish, you literally 'do a mistake', which is göra ett misstag.

If I wanted to say 'I had NOT made a mistake', where would the word inte go in this sentence?