Learn Turkish: Pil tasarrufu için ekranı olabildiğince kıs. - Dim the screen as much as possible to save battery.

It follows the structure from this lesson: the root ol- (to be/happen) + -abil (ability) + -diğ (the -dIk participle) + -i (3rd person possessive) + -nce (adverbial suffix of extent). Literally, it translates to 'to the extent that it is able to be'.

The lesson also teaches using kadar. Could I say olabildiği kadar here?

Yes! Olabildiğince and olabildiği kadar mean the exact same thing in this context ('as much as possible') and you can use either one interchangeably.

Doesn't the verb kısmak usually mean 'to turn down'? How does it mean 'dim'?

Kısmak broadly means 'to reduce,' 'to restrict,' or 'to turn down.' Just as you would use it to turn down the volume on a TV (sesi kısmak), you use it to turn down the brightness of a screen (ekranı kısmak), which translates as 'dimming' in English.

Why is it pil tasarrufu and not pilin tasarrufu?

Pil tasarrufu is an indefinite noun compound, meaning the first word (pil) just tells us the type of saving. If you used the genitive case and said pilin tasarrufu, it would mean 'the battery's saving' (as if the battery itself is the one choosing to save something!).