My best advise is to use Longman's Dictionary of Contemporary English. Each of those words/phrases has very similar meaning, each with a slightly different emphasis. "Turn into" is very general in meaning, simply to change something into another thing. For example, a knitter can turn yarn into a sweater. "Transform" carries the idea of changing something completely, making it something entirely new. For example, in "Cinderella," the fairy godmother transformed a pumpkin into a carriage. "Convert" carries the idea of changing something into another similar thing. For example, converting dollars to euros.