I found the above on the compatibility list for my TS-473 so I bought the one in the picture, however, the slot it goes in is much smaller. What am I doing wrong?
It will fit, assuming the PCIe slot is open-ended.
I’ve checked some images online, and it seems the slots in that model NAS are open-ended.
Insert the card (line-up the key-notch), and leave the rest of the card-edge dangling in space.
The card will only be able to talk with the mainboard at a lower bandwidth though. 
The second (lower) slot of the TFS-x73A series have one open ended slot and one slot that is not open ended. The top slot is open ended and your GPU card should fit in that. This is the same NAS series I have (TFS-873A) and when I had a GPU card in that model, it had to go in the top slot only. I "hacked’ things and remove the cooling fan from my GPU and made it fit into the bottom slot but it was not entirely secure. It worked, but I ended up burning up that GPU as something went awry when trying to do some video transcoding and something hung up that I was unaware of. I could keep the card cool by playing with the fan speed on the NAS, but that gets risky with the GPU.
I’ve since removed the GPU and put my fiber NIC back in place as there is not much with QNAP that a GPU buys for you. Yes, you can use Hybrid DeskStation but the apps there generally suck (don’t get me started on the video player part of Hybrid Deskstation which uses Kodi - not sure how anyone can use that horrible interface). You can also use the GPU for Linux Station or your VMs but it’s not worth having it for that. Beyond that, the only “AI” stuff that the GPU helps with is QVRSmart Search.
But really the GPU does very little for you.