FOR SEVERAL YEARS, beginning when I was 11 or 12, I wanted a pet iguana badly. In my favorite comic strip, Bill Amend’s FoxTrot, 10-year-old Jason Fox had an iguana named Quincy, who looked like a cousin of the cute Nintendo character Yoshi. I knew that real-life iguanas were more dinosaur than friendly cartoon, but in the abstract I liked the casual threat implied by an iguana’s size. My mother, however, was skeptical about my ability to care for such a creature — “Where would we keep him?” she asked. “What would we feed him?” — and the matter was dropped. Years later, I met a high school friend’s iguana, the first I had ever seen outside a pet store. It was three feet long and looked cramped even in its large, screened-in pen. The iguana had an evasive, slithery gaze and a tendency to turn its head when I tried to see its face. When I asked my friend what its name was, he smirked. “I call it the iguana,” he said, as if it would be folly to give something so ineffable a name.

The iguana in Diana Wagman’s latest novel The Care and Feeding of Exotic Pets has a name — Cookie — and an entire kitchen as his habitat, which is a necessity, because Cookie is eight feet long. But despite these more luxe bona fides, the impetus of Wagman’s novel is that same iguana amor brujo I once felt: a marriage of the adult desire for control and the morbid child’s fascination with monsters. There are many monsters in Wagman’s novel, but only one is a lizard; the rest are human grotesques whose bestial charm drives the book’s suspenseful, gimlet-eyed drama. The novel reiterates my mother’s reasonable questions: how do we care for impossible creatures? With what do we nourish and sustain them?

Set on a single November day in Los Angeles, where the sky is “vacant as a starlet’s smile,” The Care and Feeding of Exotic Pets chronicles the frantic adventures of one Winnie Parker, a divorced 38-year-old suffering through a long spell of depression. It has been almost four years since Winnie’s beloved husband Jonathan left her for a younger woman. In the Echo Park bungalow where she spent the happiest days of her bygone marriage, Winnie now lives with only her 16-year-old daughter, Lacy, who is caught at that dangerous teenage intersection of hostility, entitlement, and low self-esteem. Occasionally she fields batty phone calls from her mother, Daisy Juniper, a legendary movie star who had her only child at 16 and “has been committed twice, but for insanity not drugs.” Winnie’s friends are vaguely referred to but never seen, and her days are spent caring for a dog she dislikes, working a meaningless secretarial job, and going through the motions of recreation with no real enjoyment; on the morning that the novel begins, she is dressed in unwashed whites for a tennis lesson that seems to her about as appealing as a day at the DMV.

Winne is badly used by everyone she loves, from Daisy, whose idea of a consoling phrase is “We can’t all be important,” to Lacy, who says of Winnie, “Mom just works in an office. She’s nothing.” She feels unable to get over Jonathan, whose 23-year-old second wife Jessica possesses a pneumatic rack: “Lacy said Jessica’s boobs were fake, but Winnie thought they were just fresh and unused.” In contrast, Winnie feels “deflated, like an old balloon caught on a fencepos...