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A more general version of this statement was shown by Kimura in Acta Arith. (2003). His corollary gives $\gg \sqrt{X}/\log X$ such fields ${\Bbb Q}(\sqrt{-d})$ with $d\le X$, and also allows you to add further splitting conditions. There is an extensive literature on divisibility and indivisibility of class numbers and Kimura's paper has many other relevant references.