It is the easiest question to answer: you don’t hold the election in May. You hold it either on the primary day (this year, September 6), or during the general election.
Now, I know that education activists/advocates and those involved with the School Boards up and down our state object to that for two reasons: 1) it would politicize School Board races and make them partisan and 2) it would drown out attention to School Board races and the candidates in them when they have to compete with the bigger names on the ballot, like Governor, Senator, and President.
These are valid concerns, but the desire to increase turnout and the desire to keep school board elections non-partisan are mutually exclusive. You can’t have both. Right now, only the truly invested and knowledgeable education advocates and parents and teachers are voting in these elections. And in some ways, that’s good, even if it is an extremely low turnout. You don’t have uninformed idiot voters electing a candidate for non-education-related reasons.