Some might argue that he had notable roles before that, but I think most people will argue that his career really kicked off after those two movies in particular.
EDIT: According to Wikipedia, "he rose to fame with films such as Coming to America (1988), Goodfellas (1990), Patriot Games (1992), Juice (1992), Menace II Society (1993), True Romance (1993), Jurassic Park (1993), and Fresh (1994)", which means that he was exactly-ish 40 years old when Coming to America (1988) was released.