As the other members have suggested/explained, you can't make a contraction with the verb 'want'.
In modern-day English, the only verbs which make their negative forms with 'not' - and therefore form contractions with 'not' - are: 'to be' (isn't, aren't, wasn't, weren't), 'to have' and 'to do' when they are auxiliary verbs (eg I haven't seen/ I didn't see), and modal verbs (can/could, will/would, shall/should, may/might, must, ought).
All other verbs use an auxiliary to make the negative, so the word 'not' forms a contraction with the auxiliary verb rather than the main verb (I don't want, I didn't want).
I know this sounds a little confusing , and also rather illogical. However, the more English you see, hear and use, the more natural these forms will seem to you.