To be sure, the French translation of the English word "normal" is normal -- but the situation is more complicated than it looks. French speakers also use normal to mean "natural" or "the way things should be" or "what you would expect."
Consider these examples:
C'est normal qu'il soit fâché ! It's not surprising that he's angry. It stands to reason that he's angry. You would expect him to be angry.
Il n'est pas rentré. Ce n'est pas normal ! He hasn't come home. Something must have happened.
Ils ont encore augmenté le prix, ce n'est pas normal ! They've raised the price again; it's not right!
Je n'ai pas encore reçu le texte à traduire. Est-ce normal ? I haven't yet received the text to be translated. Is something wrong? or Has something happened? or Is everything OK?
The adverb "normally" in English is a synonym of "usually" or "habitually" but the French adverb normalement can also be a synonym of si tout va bien (if everything goes well).
Normalement, elle arrivera à 15 h. If everything goes as planned, she'll arrive at 3 pm. (Note that "normally" would make no sense here.)
Tu es libre ce week-end ? Oui, normalement. Are you free this weekend? Yes, I should be.