Yes if you are using BP this is normal. Try reducing the amount you apply onto your skin and keep on moisturizing. Everybody's skin reacts different to BP some are more sensitive than others which it looks like this may be the case here.

You can always reapply moisturizer throughout the day if your skin feels dry. Or, maybe even try to switch to a thicker one if you would like. I have had success using two different kinds...a lighter one during the day and a thicker, more emollient one at night.