Learn Croatian: Pusti me da upalim tuš! - Let me turn on the shower!

In Croatian, you can use the infinitive after another verb only if both verbs share the same subject (e.g., Želim upaliti tuš - I want to turn on the shower). Here, the subjects are different: you are letting, and I am turning on. When subjects differ, you must use da + present tense.

I thought upaliti meant 'to light' or 'to ignite'. Can you use it for water?

Literally, yes, it means to light a fire or turn on an electronic device. In everyday colloquial speech, people frequently use upaliti for turning on the shower (tuš), even though pustiti vodu (to let the water run) or otvoriti vodu (to open the water) would be more technically correct for plumbing.

Could I say Neka upalim tuš instead of Pusti me da upalim tuš?

Not quite. Neka + present translates more to 'Let me...' in the sense of a general wish or allowing a situation. Pusti me is a direct imperative to another person, actively telling them to step aside, give you permission, or release you so you can do the action.

Why does tuš look like it is in the nominative case here?

It is actually in the accusative case because it is the direct object of upalim. However, tuš is a masculine inanimate noun. For masculine non-living things, the accusative form is identical to the nominative form.