Sounds like a bike fit issue, though technique may play into it a little too.

Do you have elbow injuries/issues in other sports?

I have no idea what would cause elbow pain, I haven't read of too many others who have experienced it (so it's definitely not "a normal thing", even for new riders), but I would look into maybe your handlebar width? If it is radically too wide (more likely) or too narrow (less likely, but possible), it could cause you to grip the bars at an angle that puts torque on your elbow... perhaps? When you grasp the bar, your wrists should be more or less straight in all planes, and your shoulders, elbows and wrists should roughly align (some people prefer a bar slightly wider, so your grip will be a bit wider than your shoulders, but it shouldn't but LOTS wider -- or narrower...). And you should have some bend in your elbow -- your arms should be fairly "loose." Does this sound like your position?