Unless we are talking something like speedway motorcycling[1] a constant-rate turn is accomplished by, after the initial counter-steering, pointing the wheel in the direction of the turn, such that the self-uprighting tendency of the bicycle matches the speed at which it falls to the ground due to gravity.

If you kept pointing the front wheel to the other side of the turn, the bike would fall over. (This property is what you use to initiate the turn, but not to maintain it.)

[1]: https://3.bp.blogspot.com/-xTxeag2HCjs/WsXXrKRxWrI/AAAAAAAAc...