普通 is a more common, everyday word for 'usually' or 'normally'. 通常の場合 is more formal and often used in written or official contexts.
一般的に is an adverb meaning 'generally' or 'in general', while 通常の場合 is a noun phrase that can also function adverbially, emphasizing the 'usual case'.
Compound of 通常 (つうじょう, 'usual, ordinary') + の (possessive particle) + 場合 (ばあい, 'case, situation').