The Karate Kid Part III was released on June 30, 1989. This is what a typical advertisement for the movie would have looked like in your local paper. You can see an ad for Star Trek V: The Final Frontier below it.
This particular weekend, June 30-July 2, was exactly one week after the June 23 release of Tim Burton’s first Batman movie. Below is the top 10 box office results for the weekend. The Karate Kid Part III finished third after Batman and Honey, I Shrunk the Kids.
Other notable sequels that week include Ghostbusters II, Indiana Jones and the Last Crusade and the aforementioned Star Trek V.