States that declared themselves refuges for transgender people have essentially issued an invitation: Get your gender-affirming health care here without fearing prosecution at home.

Now that bans on such care for minors are taking effect around the country — Texas could be next, depending on the outcome of a court hearing this week — patients and their families are testing clinics' capacity. Already-long waiting lists are growing, yet there are only so many providers of gender-affirming care and only so many patients they can see in a day.

For those refuge states — so far, California,...