what's the fastest turnaround time to make a passport for my baby?

Once you've gotten all the needed documents in and payment done, you will be notified via post to collect your passport once it is ready. A pink acknowledgement card will be mailed to you within 1 week from the date ICA receives your application. So I reckon you can get it in one week if conditions are right. But if you take into account the peak season (right before June and December holidays), it might take longer so make sure you get your applications in early.