is it normal for your lips to bruise when you have a coldsore?

No cold sores don't cause bruising. They cause blisters or open sores on the lips that itch, burn and or tingle.
If it is a cold sore it may take about a week or two to go away with treatment.
You may want to get this checked out by a doctor to be sure if it's a cold sore or some thing else.