No, it is not normal to get bruises just by drying yourself off with a towel. But if you are literally "whipping" yourself with the towel, then yes, you can get bruises.
If you are concerned about easily bruising, then make an appt. with your PCP.
A patch of red dots is not a bruise.
You may have an allergy to the soap that you are using. Only use soaps that have the same pH levels as our skin and those that have no perfume. Baby washes will be fine.
You may have a condition called Rosacea which can be triggered by hot water.
Make an appointment to see your doctor and ask for a swab to be taken of your skin to identify if there are any fungus problems on your skin that may need treating. Washing does not get rid of fungus.
Your doctor may know what it is by looking at the skin and if he suspects that it may be fungal problem will prescribe you with the appropriate medication.