Many site issues can be caused by corrupt cookies or cache.

Clear the Cache
Firefox Tools > Options > Advanced > Network > Cached Web Content: "Clear Now"

and

Remove Cookies
Firefox Tools > Options > Privacy.
Under History, select Firefox will Use Custom Settings.
There is a button on the right side called Show Cookies.

If there is still a problem, Start Firefox in Safe Mode
While you are in safe mode; Firefox Options > Advanced > General.
Look for and turn off Use Hardware Acceleration.
Poke around safe web sites and see if there is still a problem. Then restart.

Can you provide a URL of one of those problematic sites? Not your statement page but somewhere else anyone can access.

This sounds like an issue with the built-in PDF Viewer.

The latest stable and latest development versions of the built-in PDF viewer are available as an extension and these version can possibly perform better than the version shipped with the current Firefox release.

Latest development version of the PDF Viewer (pdfjs) extension:

• http://mozilla.github.io/pdf.js/extensions/firefox/pdf.js.xpi

See also:

• https://support.mozilla.org/kb/disable-built-pdf-viewer-and-use-another-viewer

Unfortunately, I've only noticed the issue on two sites, my bank and the Unemployment office, where I click on their Javascript form to bring up a calendar to click on to enter a date when adding places I've applied to. That's only available once a week. I haven't noticed it anywhere else. I'm sure there are others, but I will have to keep up with it and see if it happens anywhere else.

Thanks, but I don't think is issue is PDF related as the site doesn't send a PDF. Good thought, based on the limited information I provided, but not the issue.

Thank you all for trying to help. None of the suggestions seem to have worked, yet. I'll keep looking.

So, Firefox Safe Mode also didn't help at all?

Nope, but I did a bunch of things that first broke and then fixed the problems. I have no idea how I did it.