I tested Lazarus trunk on Linux Mint 18.2, both GTK2 and QT: no issues.
I can type as quick as I want, no "jumpyness".
Maybe attach an OnKeyDown event and see if the Key there is what you expect?
The "cursor" by design only advances when you type a character or use the arrow keys.
The moving of the cursor happens in SelectNextChar and SelectPrevChar.
Put some debugln statements there (and/or in InsertChar) to monitor what you want.
Using the debugger to step won't do any good, since this will force you to type "slow".
Bart