Is it sufficient to perform ritual bathing (ghusl) after menstruation or seminal discharge by only putting one's head under the shower once, while running fingers through the hair to ensure water reaches the scalp, instead of pouring three handfuls of water over the head, especially if the hair is long and thick?

The obligation in ghusl is to ensure water reaches all hair and skin, including the roots of the hair. As for pouring three scoops on the head, that is recommended (mustahabb), not obligatory. So, if water reaches what it is obligated to reach by any means and in any number, the purpose has been achieved, and one is absolved of the obligatory ghusl. Based on this, this ghusl is undoubtedly valid.