Learn Turkish: Yeni bir bilgisayara ihtiyaç var. - There is a need for a new computer.

The word ihtiyaç (need) pairs with the dative case suffix -(y)a or -(y)e on the thing that is needed. In English, we say a need 'for' a computer, but Turkish phrases it like a need 'to' a computer. That is why bilgisayar gets the -a suffix to become bilgisayara.

Why is the word order Yeni bir rather than Bir yeni?

In Turkish, when a noun has both an adjective (like yeni) and the indefinite article bir, the adjective almost always comes first. So 'a new computer' is phrased as yeni bir bilgisayar (literally 'new a computer').

How would I change this to say 'I need a new computer'?

You would add the possessive ending for 'my' to the word ihtiyaç. When you do this, the final ç softens to a c, making it ihtiyacım (my need). The sentence becomes Yeni bir bilgisayara ihtiyacım var (literally: To a new computer, my need exists).

Could I use the word lazım for this sentence?

Yes! You could say Yeni bir bilgisayar lazım. Both express necessity, but notice that with lazım, you drop the dative case -a on the computer. It literally translates to 'A new computer is necessary'.