Under the teacher's guardianship, the young artist gradually found his own style.

guardianship 是由名词 guardian（监护人、保护者）加上后缀 -ship（表示状态、身份或关系）构成的名词。guardian 来自动词 guard（保护、看守），所以 guardianship 字面意思是"监护的状态"，引申为"指导、保护"的意思。