The random selection of tense in these translations is problematic, without some kind of time marker it is impossible to say whether this is current or past. 'I think that she is a woman' should be acceptable. Also perempuan can be translated as girl so 'i think that she is a girl' should be accepted too.
'bahwa' is used to introduce a clause.
The clause that follows 'bahwa' usually functions as the object [O] or complement [Pel] of the sentence.
-1) 'Saya [S] berpikir [P] (bahwa) dia perempuan [Pel].'
'bahwa' is optional in this sentence.
You can also omit 'bahwa' and rephrase the clause like a "quote":
-2) 'Saya berpikir: "Dia perempuan" '
'bahwa' can also be used as a modifier, like this:
-3) 'Berita itu [S] mengherankan [P] kami [O].'
-4) 'Berita bahwa orang itu bukan perempuan [S] mengherankan [P] kami [O].'
'The news that that person is not a woman surprised us.'
In a structure like in sentence 4), the bahwa clause is part of the subject.
The clause modifies the head noun of the subject ('berita').
It gives extra info about the head noun ('the news').
'bahwa' is mandatory in sentence 4).
In a sentence like this, you can only omit 'bahwa' if you also omit the clause.
What remains is sentence 3).
In all the examples above, 'yang' cannot be used to replace 'bahwa'.
'yang' can also be used as a head noun, as a modifier, or to introduce a clause.
'yang' functions differently, examples:
-5) 'Saya berpikir bahwa dia perempuan yang suka balapan kuda.'
'I thought that she was a woman that likes horse racing.'
'yang' clause is modifying 'perempuan'.
-6) 'Yang mengherankan kami adalah berita tentang perempuan itu.'
'What surprised us was the news about that woman.'
'yang' is the head noun and the subject.
-7) 'Berita yang mengherankan kami adalah berita tentang perempuan itu.'
'The news that surprised us was the news about that woman.'
'yang' modifies the head noun 'berita'.
These are some uses of 'bahwa' and 'yang'.
'yang' can be used in other ways as well.
I think it's one of the most frequently used words in Indonesian.