Learn English – Which form of verbs should be used after conjuction words in interrogative statements

All full questions* require a finite auxiliary verb^† (that is, an auxiliary verb tensed for either past or present) before the subject. When the verb in the sentence is a construction with an auxiliary, this is accomplished by switching the subject and the auxiliary. When the verb is a lexical verb with no auxiliary, it is accomplished with DO-support: the tense is removed from the verb and put on a form of DO before the subject:

He is John Smith. → Is he John Smith?
He is [who]. → Who is he?

He has gone to town. → Has he gone to town?
He has gone [where]. → Where has he gone?

You look like an executive. → Do you look like an executive?
You look like [what]. → What do you look like?

Your second sentence, however, confuses two different constructions. There What is not an interrogative, a question word, but a relative pronoun heading a clause which acts as a noun phrase. The statement form would be:

[_Subject What you said] [_Verb turned out] [_Complement to be true].

Consequently, the question form would be:

Did what you said turn out to be true?

* As Peter Flom points out, questions may be truncated very substantially in discourse context.
^† BE is always considered an auxiliary, even when it is the only verb.

An open question is one that has many possible answers: subject or object questions are examples of this, as the there are multiple possibilities.

A closed question requires a yes or no answer.

Closed questions in the past simple are usually asked using **did*.

Did you break it?

I hadn't thought about it before, but I think that you have identified a pattern, in that we also use did for object questions, presumably so that the object pronoun can be moved to the start of the sentence.