Moving to the right Subsubsection in TOC

First a couple of comments. (a) If you will be submitting your document to the AMS do not change any of the typesetting. (b) I don't really understand your question; you ask about the ToC but show a desired result in the document body.

The generally accepted method of changing the appearance of ToC entries is to use the tocloft package, which doesn't seem to work with the amsbook class. Below shows how it can be used with the book class to give your desired extra indentation for subsubsections in the ToC.

% subsubsecprob.tex  SE 567344
%documentclass{amsbook}
documentclass{book}

setcounter{tocdepth}{4}
setcounter{secnumdepth}{4}
usepackage{tocloft}
addtolength{cftsubsubsecindent}{5mm}

begin{document}
tableofcontents
chapter{Chapter}
section{Section}
subsection{Subsection}
subsubsection{Subsubsection}

end{document}

I have no intention of trying to make tocloft compatible with ams..., partly because I may not be capable any more, but principally because the AMS have their definite opinions as to how their documents should be typeset. --- GOM