The race cars are to fast to learn how to H shift , IMO .
Start with slow uf1 or similar,you will have loads more time during braking to downshift /upshifts on exit
Don't use the shifter to fast or hard ,learn where the gates are gently.
Blipping the throttle to match revs on downshift helps , but the pedals are not really set up for this out the box,the brake is to soft to proper heel/toe and not close enough to throttle.
Your totally correct about the lag of g-forces ,when messing up a change, your not getting your face slammed against the windscreen to tell you it was messed up .
I found using 'Acceleration shift viewpoint ' settings useful.
Now I use H-shifter for messing around on road cars ,and paddles/auto clutch for racing ,since I don't have a force dynamics rig to give me G
Try a realistic steering wheel angle / lock on front wheels ,
as per this thread ,to get real life steering ratio .
Lock on front wheels makes a huge difference to sensitivity.
In the FZR I use 540 steering/21 wheels to give a 12.8:1 ratio .
If you use 25 at the wheel the ratio goes to 10.8:1, much more twitchy.
Dont try and be as fast right away , it will take time .
Patience is a virtue