Is 青少年自殺越來越嚴重這問題完全失控 grammatically correct?
[1/青少年] [2/自殺] [3/越來越] [4/嚴重] [5/這問題] [6/完全] [7/失控]。

[1/Young people] [2/commit suicide] [3/became more and more] [4/serious] [5/this problem] [6/completely] [7/out of control]。

The phrase "自殺越來越嚴重" (commit suicide become more and more serious)  itself is awkward, because commit suicide is a verb phrase; 嚴重 is an adjective. Adjective cannot modify verb. 

The correct phrasing is:

[1/青少年][2/自殺][3/這問題][4/越來越][5/嚴重],[6/已經][7/完全失控]。

[1/Young people][2/commit suicide][3/this problem of][4/become more and more ][5/serious],[6/already][7/out of control]。

notice:

[1/Young people] [2/commit suicide] [3/this problem of]

"** problem**" is the head noun,  "young people suicide" is the relative clause. In English grammar, the head noun is placed in front of the relative clause like so: "This problem of young people commit suicide" but in Chinese grammar, the head noun is placed after the relative clause like so: "青少年自殺這問題"

已經 (already)ties " this problem of young people commit suicide" to the follow up statement " completely out of control" together. 

if you don't add 已經. the sentence will be split in two. 

青少年自殺這問題越來越嚴重. (這問題)完全失控。

-subject (這問題), is omitted, because it is stated in the previous related sentence.

A final grammatically correct edition would be:

"青少年自殺這問題越來越嚴重, 已經完全失控。 

" This problem of young people commit suicide became more and more serious, already out of control."