I recently saw from Rav Mattisyahu Solomon (in a sefer on chinuch, where he has a long piece about libraries) that public libraries are very problematic (this is talking about non-Jewish books, almost all of which are problematic). History and science books are seemingly OK, but even these are often problematic (because they talk about evo***ion).
If a child needs a particular book, better that the parent should go to the library and take out that book for the child.