My website does not know show up on Google Search even when I google content verbatim from it

I am an academic and recently made a squarespace site for my work, and after 4 months, still the website does not show up on Google despite my checking all the standard bugs (made sure it's not private, checked that it's indeed indexed, verified it on Google Search Console etc.). It seems like something deeper is going wrong, because if I google my name, the website appears first on Bing and DuckDuckGo but is nowhere to be found on Google. In fact, even if I google my name and content verbatim from the home page, it still won't show up on the search results. And adding to the puzzle: more minor pages in the website will appear for certain keywords on the last page of Google search results, but the home website page won't appear basically ever—again, even if I google words verbatim from it. The main page only shows up when I put entire sentences in quotes or if I google the URL. Any idea what might be going on here? 

What keywords are phrases are you using and can you pass along your url?