Yes, very normal. Your husband has undermined your trust and it seems he is lying about it on top of it. In this case, he is betraying your trust twice over.
If he had affairs and stopped his behavior then made amends, you would have a chance to recover from the betrayal and rebuild your trust. But as it is, he continues to cheat and lie about it so you keep getting hurt and he doesn't take responsibility.
If your husband is willing to not only lie to you but also to others, then he is not seeing his behavior for what it is. He is in denial and is not coping with the consequences of his behavior.
I highly recommend you see a therapist to help you deal with not only how you feel about what your husband is doing, but what action you need to take regarding your marriage. It is easy to feel you just want to leave it all behind or stay and ignore his behavior, but by taking your time to sort out your feelings and make a clear choice, you leave very little to regret and you will feel more settled with your decision.
To find a therapist, talk with your doctor. Or you can also talk with your pastor. Another option is to search on line at http://therapists.psychologytoday.com/rms/.
Here are some other resources to help you:
Should I Stay Or Go? : How Controlled Separation (CS) Can Save Your Marriage by Lee Raffel
Too Good to Leave, Too Bad to Stay: A Step-by-Step Guide to Help You Decide Whether to Stay In or Get Out of Your Relationship by Mira Kirshenbaum
Surviving Infidelity: Making Decisions, Recovering from the Pain by Rona B. Subotnik and Gloria Harris
You can find these books on Amazon.com or your local library may have them for you.
I hope this has helped you,