Coming from GT, I can tell you that you're comparing the wrong games. Horizon is the "arcade" racer, Motorsport is the "sim" racer. Try Forza Motorsport 7 (FM7) if you want a better comparison to GT.
Having said that, I think you might have been onto something: I didn't know you were using an older X360 controller. There have been improvements in analog precision with newer controllers, I would recommend getting a new Xbox controller. Granted, per above, an analog stick never really meets the correct precision of a steering wheel, and the only reason a corner will feel "smooth" in a sim racing aspect with a controller is because the game is "assisting" you whether you realize it or not. It's impossible to map the wheel 1:1 with the stick and not end up in the ditch 50% of the time. In FACT, if you WANT to experience what happens when you do that, I believe Project CARS or something did 1:1 with the analog stick (so that you were literally turning the wheel the same amount your analog stick was getting pushed with full lock being all the way), and it's nigh impossible to do well.
GT, FM, FH, etc. in which they expect the core of their playerbase to be on controllers have modulation and assistance on the analog stick because it's an absolute must. This is why GT has that "lag": it's there on purpose to absorb rapid movement or you'd be whipping that "steering wheel" like a madman if it gets mapped 1:1 or doesn't assist you in some form or another. It's why I duly laugh when people start getting all serious with sim racers on a controller lol
I know it's the "arcade" racer. I'm comparing it to my experience with FH5, not GT when it comes to handling.
The one part I am comparing it to GT with is how it handles the analog input. In GT I can keep a steady turn rate with precise analog input. GT indeed has 'lag' implemented in how quickly the steering wheel responds to analog input, but you can 'switch lanes' as slow as you want without it over re-acting. If you watch my steering input in this video you can see the 'smoothing' going on. It's the red dot moving around above the rev bar.
If you flick the stick from one extreme to the other it indeed takes some time for the red dot to follow suit. Yet you can perfectly keep a steady line and make quick small adjustments (I do play on max sensitivity for quick reaction) The ref dot can be very close to the center while remaining precise and following my small input without having to 'modulate'.
With Forza Horizon it takes longer for the car to respond to any input (dead zone) and then over reacts when it does 'engage'. Hence continually correcting throughout a sweeping corner. But maybe it's the 360 controller. It's not an old one though, I bought it when FH4 came out since the game doesn't support DS4. (Maybe a DS5 works, doubt it)
And indeed, Project cars was a nightmare on the DS4.
Anyway, lvl 88 already, finished up the street races and Marathon race, ready to take on the gauntlet next (final dirt race)
Edit: Yes, probably the controller. I went to controller properties in windows and the controller itself has a big dead zone, then 'jumps' into action. Not Forza Horizon then, controller's fault :(