What's the best way to ship heavy medical supplies?

Try Flat-rate shipping. USPS has boxes that you can ship anything in for one price regardless of its weight. They aren’t especially large boxes, so you may have to use many of them, but the cost should work out to be less in the end, than shiping normally.

Kimbles is right, but as a Mailman, I HATE YOU. I better get a huge Christmas bonus.

My brother shipped some heavy bronze artwork through Greyhound. Apparently, as long as you have someone to drop it off and pick it up from the bus stations they will ship things fairly inexpensively in the baggage/cargo holds.