'Come up' means 'arise', and 'turn up' means 'arrive'. Usually, they aren't interchangeable. For example, we would normally say 'A problem has come up', meaning 'arise'. It suggests something bubbling to the surface. It's used for situations, not for people or things. 'Turn up' is used for people and things, meaning to arrive at a particular place, from another another place. We often use it when something arrives in a place, or at a time, when it is not expected. 'John turned up at the party three hours late' or 'I couldn't find my keys for ages, and then they turned up on my desk. It was very strange.' You couldn't use 'come up' here. You are right that you could use either phrasal verb in the particular sentence in your question, with almost no difference in meaning. However, in most cases, they would not have the same meaning.

And good luck with your query on WordReference Forum, onur8runo. I'm watching that page with interest. Maybe somebody there can help.