Oceanblue,
No, it is not normal. You are risking serious nerve damage by letting this continue.
Let me tell you about the problems with hand numbness I first experienced when taking up cycling three years ago. My initial problem was that the cycling store sold me a bike which was way too small for me. After riding about five miles my hands would start to get numb and, the longer I kept riding, the worse it got. On some rides by 10 or 11 miles the hand numbness was so bad that I could no longer actuate the brake lever with my right hand. Like you, I would experience numbness and tingling in my hands long after the ride ended.
To make a long story short, I got a professional bike fitting and, armed with that information, went to a different bike store where they built me a new bike (using the components from my first bike) on a frame that was 4 cm larger than the first (I went from a 52 cm frame to a 56 cm frame). That made all the difference in the world. Since then I have been able to enjoy cycling and have completed many long rides including a century ride.
I do occasionally suffer minor hand numbness. To relieve this I change the positions of my hands on the handlebars frequently. I use the information provided by DanoXYZ in the famous "Numb Hands" post (referenced above) to keep myself from falling into bad habits.
Good luck. Keep us posted on the steps you take to relieve this condition.