The media giant 21st Century Fox, the empire run by Rupert Murdoch, made an $80 billion takeover bid in recent weeks for Time Warner Inc. but was rebuffed. The bold approach could put Time Warner in play and might again ignite a reshaping of the media industry, prompting a new spate of mega-mergers among the nation’s largest entertainment companies. Mr. Murdoch has built a global media juggernaut over nearly five decades spanning studios, television channels and newspapers, in part, by pursuing bold deals that were often rebuffed at first by the targets of his overtures, only to later acquiesce. Time Warner on Wednesday confirmed that it had rejected a cash and stock offer from 21st Century Fox, saying that it was not in the company’s best interests.