Bubble-wrapped slides in a cardboard box works fine. Send it into the X-ray separately and it won't result in problems with the TSA.

The only way I can see the TSA being an issue if the slides are packed like suggested here is if a passenger is a TSA haraSSSSment target as indicated on the boarding pass at time of check-in. And even then, as long as one has some material to properly repackage the items as suggested here, there shouldn't be any issues.

It's very routine for slides to be transported even by FedEx (and sometimes even USPS) --talking huge numbers daily across the country -- where the packaging is most frequently anything but bulky. Even when such items are transported by doctors/patients/other passengers on flights, the TSA is never the problem when it comes to slides from the pathology labs; the problem tends to be with slide fragility or some other factors independent of screening by the TSA. The OP's pathology lab should be able to provide appropriate suggestions on transport conditions and perhaps even appropriate packaging material. Pathologists routinely send and receive slides by package/mail services, and it's pretty ordinary for those who are specialists in this field to carry them in carry-on or checked luggage on the way to meetings/conferences that pathologists use for CME credits or use to teach at places like the Armed Forces Institute of Pathology (since closed). The Pathologists/lab almost certainly have appropriate packing material which they could provide or suggest using; but given what they charge they may as well add in the packaging material for no additional charge.