In English, normally can only be used to refer to what is (or isn't) the usual, normal situation. Look at the following examples :
You don't normally chew gum. Why are you doing it today ?
I don't normally wear a tie to work, but today I've got an important meeting with a client.
I'm sorry about this - we don't normally have problems with the lift.
Normally cannot be used to express probability as in the French 'il devrait normalement être à l'aéroport maintenant'. In English, we would say : 'He should be at the aeroport by now'.
Equally, we say : 'You shouldn't have any difficulty finding us as we're right opposite the station.'
