I don’t think it is really very uncommon for a person’s gums to bleed. I think it really may depend on the medication, and how hard a person really brushes their teeth 🦷. So when You do don’t push so hard on your teeth and try to avoid your gums.

My mama was on dilantin for years and the same thing happened to her. Constant abscess's.

Each person is different with their medications. Bleeding gums from AED's are possible for anyone. I was on Keppra but not long enough to know if it would cause bleeding gums. I had that issue with Topiramate. I was on that one for 2+ years. I told my Neuro and he said it was possible with any AED.
For pulling infections out of your mouth, Use Organic all natural Coconut oil everyday for a couple of minutes a day for 2 weeks. Depending on the infection, the time used will very.

My daughter takes Depakote and her gums bleed.

That could be a sign of low platelets which is caused by many medications