Is it normal to have increased sciatica pain after starting a round of prednisone for bronchitis? The pain seems to be nerve pain.

No . Likely you have something going on in your spine that is unrelated to the prednisone usage. You could have been coughing with bronchitis? If so, coughing is known to cause a herniated disk resulting in the sciatica.
Coughing. I would suspect that the coughing during a case of bronchitis could actually exacerbate the sciatica. It is like coincidental that the pain increased after starting the steroids.