Your plan of Googling a phrase (with quotes around it) is good thinking, but a sentence like "The place where she is now is closer than what you might think" is too long and too specific to give an accurate representation of the phrase's commonality or correctness .
Breaking the sentence into smaller sections and removing words that are unnecessary to the meaning of the sentence (such as "what" or "might" in this case) will produce more results.
Also, experiment with Google's verbatim feature to return different search results. It's #2 on this list of Google Search tips. I use the feature constantly, especially with technical searches like coding questions.
Another Google tool that is informative and interesting is the Google Ngram Viewer, which search through millions of books (some written centuries ago) to see & compare word or phrase frequency over time. Phrases don't use quotes; they are separated by commas.
Here is an example of Ngram, breaking up your example phrase into 2 parts. Note the differences when you change the corpus (like
British English versus
As for the phrase in question:
The place where she is now is closer than what you might think
...make sense, but sounds slightly unnatural to me. I don't think what belongs in it, and personally I would add a comma. I would write it:
The place she is now, is closer than you might think.
I suspect my comma placement is probably not "correct" but it makes sense to me (and besides, the very definition of "correct English" is rapidly changing). As you can tell, I tend to write with lots of quotes, commas and italic, especially online like this, because I feel that it helps convey my "intended tone" by breaking up sentences with commas, and bringing attention to certain words.
Unless you're planning on becoming an English Teacher, perfect grammar is absolutely unnecessary. The important part is that people know what you're trying to communicate (even if it's not the same way others would spell or phrase it).
The idiot president of the USA has terrible grammar and likes to make up words to suit his needs, but his bank account hasn't been negatively impacted. :)