I enabled the on-screen keyboard and recorded myself typing into the address bar
Then in iMovie I reduced clip speed to 1% and found it takes 8.4s for the first keystroke (once recorded by the onscreen keyboard) to show up in the address bar
If I'm doing the math correctly, that means there was an 8.4s/100 = 0.084s = 84ms input lag!
Then for the next few keystrokes:
13s > 130ms
11s > 110ms
0s > 0ms
17s > 170ms (but two chars showed up at once)
Same but in safari:
So there is definitely something going on here.