You haven't provided full mwe, so I expanded your sample to illustrate the problem:
\documentclass{article}
\begin{document}
\begin{figure}
\begin{minipage}{0.6\textwidth}
\makebox[0.4\columnwidth][l]{line1}
\par
\makebox[0.4\columnwidth][l]{line2}
\par
\makebox[0.4\columnwidth][l]{line3}
\end{minipage}
\begin{minipage}{0.375\textwidth}
\centering
%\includegraphics{pic.jpg}
image
\end{minipage}
\end{figure}
So what's the problem?
\end{document}

The problem is that two minipages which should be each next to the other are stacked down, and following text floats to the right column, reserved for the minípage. The first problem is with margins, which were provided by linked question's OP, which causes minipages to overflow. We can remove the margins and minipages will float correctly. The other problem is with figure environment, we need to provide css declaration to support enclosed floating objects. Corrected .cfg file:
\Preamble{xhtml}
\makeatletter
% to strip fraction from \textwidth
\def\striptextwidth#1\textwidth{#1}
% we must refer to minipage from the css file, because tags are beeing
% written before we know dimensions
\newcount\mini@count
% save original minipage
\let\oldiimini\@iiiminipage
% redefine minipage
\def\@iiiminipage#1#2[#3]#4{%
% calculate dimensions and save it to macro
\xdef\miniwidth{\strip@pt\dimexpr(\striptextwidth#4pt)*100\relax\%}
\oldiimini{#1}{#2}[#3]{#4}
}
\ConfigureEnv{minipage}{\advance\mini@count by 1\relax\ifvmode\IgnorePar\fi\EndP\HCode{<div class="minipage" align="center" id="minipage\the\mini@count" style="border:1px solid black;">}}
{\ifvmode\IgnorePar\fi\EndP\HCode{</div>\Hnewline}%
% we must write dimension here to the css file
\Css{\#minipage\the\mini@count{width:\miniwidth;}}%
}{}{}
\makeatother
\Css{div.minipage {
float: left;
}
}
\Css{div.minipage:last-child {
clear: none;
float: right;
}
}
\Css{div.figure{clear:both;overflow:auto;width:100\%;}}
\begin{document}
\EndPreamble
Notice that I changed changed Preamble to xhtml, removed HtmlPar configuration which is unnecessary and incorrect, removed margin declarations in .minipage css and added css for div,figure. The result is much better now:
