Can irrational fear be a side effect of experiencing death in the family?

A consequence?, yes: Especially for very sensitive people, an intense or traumatic event or loss can have major powerful consequences. Also, fears can be rational or not and can range from manageable to immobilizing. But yes to your question. And, you can find help in grief counseling or psychotherapy. If help is also needed for underlying fragility, medication might be added. My sympathy ; hope for relief.