Batch export PSB files to .tiff and .jpeg

You can also prepare simple Action. Just go to Action Panel - Window > Actions

Open uour PSB file

Create new action and recodrd three steps.  Start with save as Tiff - be sure not tu change name

then save as JPEG - also dont change name and finally close PSB image. Stop recording Action

Now you can celect all files in Bridge and go to Tools > Photoshop > Batch command and perform your just prepared action

The Image Processor script that ships with Photoshop can work with PSB files and save to both JPEG and TIFF.

The third party Image Processor Pro script can also do the same while offering more options:

https://sourceforge.net/projects/ps-scripts/files/Image%20Processor%20Pro/v3_2%20betas/

EDIT: I forgot to mention that if select PSB files in Bridge, then use the Bridge Tools > Photoshop > Image Processor menu, the files will load into the IP script (this script is available in both Bridge and Photoshop).

The Image Processor Pro script is not available in Bridge from a menu, however, if you preselect files in Bridge, then go to Photoshop and use File > Automate > Image Processor Pro, then you can select Bridge as a source rather than Folder.