Welcome to HealthcareMagic forum .
Yes there are chances that there is a local infection in the ear lobe.
Otherwise it doesn't take so long for the wound to heal.
Another probability is that of repititive trauma or handling making it difficult for wound to heal.
I appreciate your efforts to keep the region clean and recommend to use an antibacterial ointment like mupirocin
etc on the wound .Try to avoid using hydrogen peroxide
and just cleaning with an antiseptic solution will do . Ideally removing the earring and replacing it with an alternative thin string wound allow better healing and prevent closure at the same time.In case bleeding persists beyond a week do seek prompt consultation with your doctor to rule out significant causes like nutritional deficiencies or bleeding tendencies and allergic reactions.
Wishing your child good health .