Is it normal for a doctor not to do a throat culture when you have an infection? my doctor said he'll do one if the antibiotics don't work,so is it?

Not so unusual : Some throat infections/pharyngitis have a very familiar etiology and a culture is not always required to confirm the diagnosis. Hope this helps!

Null and void: I will tell you truthfully: doing a bacterial culture swab AFTER taking antibiotics does no good, as if you had had a bacterial infection when you saw him/her, the antibiotic would have already killed the bacteria, making a culture after the medicine highly unlikely to come back with bacteria growing. I personally swab before giving Abx if I think the infection could be bacterial for this reason.

Culture : Sometimes it is easy to rule out a bacterial infection with accuracy by history and physical alone, and you don't need a throat culture. It is always a good idea to check a throat culture prior to prescribing antibiotics though.