The problem in both cases is that
\titleclass{\subsubsection}{straight}[\subsection]tries to move section levels under \subsection down before inserting the new \subsubsection level. A way to work around this is to define a level with a new name, e.g. ssubsection, and then put \subsubsection to be a copy of \ssubsection; because of optional arguments etc. you will need to use \LetLtxMacro from the package of the same name to do this:
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\documentclass{tufte-book}\usepackage{letltxmacro}\titleclass{\ssubsection}{straight}[\subsection]\newcounter{ssubsection}\titleformat{\ssubsection}{% \normalfont\normalsize\bfseries\selectfont%}{\thessubsection}{1em}{}[]\titlespacing{\ssubsection}{0pt}{*3.25}{*1.5}\LetLtxMacro\subsubsection\ssubsection\begin{document} \subsection{Bar} Text. \subsubsection{Foo} Text.\end{document} 