If there is ......are unaware of為什麼have卻是擺在that的前面呢？

"There" operates as a dummy subject in the construction "there is ...", indicating that something or someone exists:
There is additional information (that) Turkish authorities have.
(= Turkish authorities have additional information.)

... have "that" we are unaware of ...
這裡的have放在that前面的用途是什麼？
不相關，如果你明白箇中道理，下次你就不會這樣問.

The "that" is a relative pronoun introducing a restrictive relative clause to further modify (restrict) the noun phrase "additional information that Turkish authorities have".
( 指那些我們不知情的額外資料)

Better way of saying: (as using "there is" is confusing)
If Turkish authorities have (any) additional information that we are unaware of , ...[main clause].

例1. [If there is additional information]：主要子句c {[Turkish authorities have(動詞v)]：(修飾語a) [that we are unaware of]：(名詞子句n)}：(修飾語A). ←← 例0.5 If there is additional information Turkish authorities have it. →→ 例1. If there is additional information Turkish authorities have that we are unaware of.
# have(動詞v)修飾[that we are unaware of]：(名詞子句n、受詞)。
* 例1.的“of”有無的差異，我就不懂；我查閱辭典的很多例句，只有一例句有“of”，其餘皆無。

The reason for putting Have----->that;
If there is additional information (that)-----rel.pron. used in defining clause introducing adv.cl
Turkish authorities have (that)-------conjunction introducing a n.cl.
we are unaware(d) of,----past participle
"we should prosecute (vt)"------main clause.