If you watched Julia Roberts cycle through Bali’s artfully cultivated rice terraces in the film “Eat, Pray, Love,” it’s easy to understand why this so-called “Island of the Gods” is often referred to as a tropical paradise. But the film, and the Elizabeth Gilbert book of the same name, obscure an unpleasant reality that has been brewing for years: Bali is struggling to cope with its own popularity.

The island of four million hosted a record 13 million visitors (nearly five million international, and more than eight million domestic) in 2016 and is on pace to better that mark this year. I wasn’t aware of this when I booked a midsummer three-week trip to Bali, but we felt the reality of those numbers immediately in Ubud, the once sleepy artist’s village where Ms. Gilbert (played by Ms. Roberts in the film) found love.

The raves I had read about Bali’s charming people and their distinctive Hindu culture were true but I wasn’t prepared for the intense traffic or how unpleasant it would be to traverse Ubud’s congested streets on foot with our boys, Leo, 8, and James, 6. Bali has more than three million registered vehicles, but we discovered that there are a host of nearby islands — Nusa Penida, Nusa Lembongan and the Gili Islands — that are car and scooter free. (Nusa means island in Balinese; Gili means small island in Bahasa Indonesian.)

The Indonesian archipelago has more than 17,000 islands, and I wanted to spend a week in one of the car-free ones. I eliminated Penida and Lembongan, since they are only a half-hour ferry ride from Bali and thus probably too accessible. Of the three main Gili Islands, each about two to three hours from Bali by speedboat ferry (or a much quicker ferry ride from Lombok), Trawangan, often simply called Gili T, is considered the party island, Meno the quiet one, and Air somewhere in between. Hoping to find our sweet spot, I booked a ferry passage to Gili Air and hoped for the best.