Learn Swedish: Vi svänger i kurvan så sent som möjligt. - We turn in the curve as late as possible.

Because it describes how the action of turning (svänger) is done, which makes it an adverb. In Swedish, you usually add a -t to an adjective to turn it into an adverb. So sen (the adjective "late") becomes sent (the adverb "late").

Why is the word for "possible" written as möjligt rather than möjlig?

The phrase så [adjective/adverb] som möjligt is a fixed construction. The word for "possible" will always take the neuter/adverbial -t ending here. It is always möjligt, regardless of what the rest of the sentence looks like.

Why do we use the preposition i for i kurvan? Wouldn't på make more sense?

In English you might say you are "on" a bend or curve, but Swedish uses i (in). You can think of it as being actively inside the physical space of the turn as you drive or walk through it.

Can the verb svänger mean anything besides turning a vehicle?

Yes, while svänger means "turns" (like going around a corner), it also means "swings" (like a pendulum, a swinging door, or swinging your arms). In this sentence, the context of kurvan (the curve/bend) makes it clear that we are changing direction.