I would not call that normal.
In a loving relationship, being open and honest means being vulnerable to the possibility of being hurt.
Some people might hold back certain bits of information if they think another might reject them because of it - for instance, that they used to have a lot of unprotected casual sex. Or that they were adopted, don't know their biological parents and don't know if they carry any genetic defects that could come out in their offspring. Or that they used to bully certain kids in primary school... etc. Those facts could have serious consequences if you're hoping for a long term relationship.
If someone isn't open about those things, it's worth considering that perhaps they are not trustworthy.
Conversely, you should feel comfortable sharing the same kinds of things with anyone you think has the potential to become a life partner.
It's a natural part of the courtship process - or should be - because honesty is the most crucial aspect of any functional and successful relationship.
On the other hand, trust in the other person's integrity and honesty is also important.
Distrust can cause the paranoid one to try to control. They demand free and random access to their partner's phone and computer, their financial records, laundry, possessions, etc. They want their partner to account for every minute of their time. That kind of behaviour destroys relationships.
Many people have a few things they keep private or secret. Usually the reason has to do with ego or shame over trivial things like mistakes they've made in the past. When someone has learned from those errors and become a better person because of it, such secrets are not a problem. They cause no harm.
When you get to know someone new, the unveiling of selves to each other can be gradual.
Do they do what they say they will do? (Keep their word.)
Do they exaggerate or minimise for the sake of a good story? Or do they tell it like it is?
Do they actions, words and tone of voice match or not? For instance, does he say everything's fine but look nervous - if so, what's the reason? It might be just be nervousness about whether you accept him or not.
Everything in relationships is about balance.
You can get plenty of info at Psychology Today - a free online magazine full of blogs by specialists.