The example you've given represents the most opposite kind of behavior possible on corner entry, because of its welded rear differential and giant cross ply tyres that will only snap on you if forcefully unsettle the car to fight the natural under-steer. OTH, in these GT cars many things will contribute to a snap if not dealt with earlier.
As pointed above, the sequential gearbox present on these cars will only auto lift on gear up (iirc) but certainly not auto blip on gear down (unless you enable the in-game assistance), and these are 2004 tyres. The approach to a corner must be different from a modern GT, try to depend less on trail-braking deep into a corner and do most of your heavy braking earlier and in a straight line, after this, just fighting the under-steer on exit will be easier on you, if not boring, because of the patience it'll require to finally go full throttle.
You'll really need to blip on 3rd and 2nd gears while keeping brakes and car balance in check; brakes are very strong for the available grip, so don't let the rear lock (fortunately setup will help). The 55 is very critical in this and probably the hardest on the pack, while the 57 is way more permissive (Shadow is well behaved too, but the MR layout can make things a bit harder). After this, you'll find them way more predictable or even understeery.
There are many setup related things you can do to make your life easier on these cars, such as softening the rear some 2 clicks, avoiding softs (more abrupt traction loss) until you get the hang of it, reducing max brake power to the minimum (you'll still have lots of braking power, this will help a lot), and moving balance a bit further to the front some 2-4% until you start cooking the front tyres to much.
More experienced players will surely help you with better tips, good luck and have fun!