Would a severe toothache be cured fast?

Yes if treatment: Is rendered. Typically severe pain is due to irreversible pulpitis or abscess. A simple filling would not be the trick, nor will medicine (analgesics or antibiotics) cure the problem. Most likely either root canal therapy or extraction will be the answer. Of course medications help with symptoms. But see a dentist asap.

Yes: Yes, an extraction of the tooth will be a fast cure. This might not be the best cure if it's a front tooth. Call a dentist to have tooth evaluated and then different treatment options.

Yes: Yes, take the infected nerve out and it will alleviate the pain.

Yes: Yes. The dentist will make a diagnosis and provide treatment which will take care of the pain.