LOS ANGELES—As Walt Disney Co., AT&T Inc.’s WarnerMedia and Apple Inc. prepare to enter the crowded streaming-entertainment market, they are racing to stand out with eye-catching shows that cost as much for a season as a big-budget movie.

These new services are hoping their planned television epics will capture the cultural conversation, like “Game of Thrones” did. They are also hoping to convince subscribers that their offerings are worth paying for in a market dominated by Netflix Inc., HBO and Hulu.

The...