Is it normal for a man to shed skin on his penis?

The most simple answer is 'no.' I first must ask whether you've been checked for any 'STD's? This is important to do. However, based upon your description, it sounds perhaps to be either a fungal infection, perhaps as a result of thrush, (Candidiasis), either on your side or your partner's.

One thing that may possibly be present as well is an itchy red rash with swelling of the glans. This may be balanitis. The term "balanitis" simply refers to inflammation of the glans. Being uncircumcised and having poor hygiene are both risk factors for this condition, but balanitis may result from other causes -- both infectious and non-infectious. It requires medical treatment.

Another thing to consider is your laundry detergent. Bio detergents can contribute to skin rashes, particularly in sensitive areas. Build up of moisture through sweating or not thoroughly drying after your bath or shower can reactivate the detergents which act as caustic irritants.

Additionally, it may arise as a consequence of dryness. Absence of lubrication during sex or masturbation can create friction.

From what you've shared, I'd suggest trying a fungal cream such as Canesten for men - a treatment for Thrush and see whether this works. If itching is present, also consider Canesten HC which contains hydrochloride.

Regardless, as mentioned at the beginning, it is always advisable for both you and your partner to have a simple check for STD's.

For anyone who is HIV positive, fungal infections can have different consequences and it's most important that professional consultation is sought immediately.