The bypass hose you have there is used when an aftermarket turbo without water cooling... HAH! It's NOT an upgrade!
Excellent. Just replace the hoses when you get the chance. You can run the turbo without the water cooling but it'll run hotter (see: efficiency loss) and won't live as long.
EDIT: The lines you have linked to there aren't the ones you need. You need the banjo couplers for the thermostat neck side... What you've posted there is just prettier version of the unused hard pipes in the photo. I'll look up the part numbers for you.
Top fitting: 90404-52033
Lower fitting: 90405-10031 and banjo bolt: 90401-14091
From your pictures though it almost looks like the bypass hose isn't running where it should... hard to tell. It should run to the back side of the thermostat tower/neck. Here's the diagram from the EPC, you're looking at the parts numbered; 16281B and 16333P.
http://www.cygnusx1.net/Supra/Librar...spx?F=1603&P=2