why does make4ht generate an extra : in this math expression?

I find that that the issue you've encountered goes away if I get rid of the specious pairs of curly braces in your code.

Here's a screenshot of the displayed html file obtained by running make4ht main.tex "mathjax". (The tex file is called main.tex. My TeX distribution is MacTeX2020.)

\documentclass{article}
\usepackage{amsmath} 
\begin{document}    
\[
W = \begin{vmatrix}
\sqrt {x} & {x}^{\frac{3}{2}} \\ 
\frac {1}{2}\frac {1}{\sqrt {x}} & \frac {3}{2} \sqrt {x}
\end{vmatrix}
\]          
\end{document}

TeX4ht passes LaTeX math verbatim to the HTML file. The problem is that the process it uses introduces lot of unwanted spaces to the output. This can cause issues in MathJax rendering. To fix this, it uses several regular expressions. One of them replaces \ { with \:{, in order to fix this issue. The problem is that it also replaces \\ { with \\:{, which causes the spurious colon. So we need to catch this case with another regular expression.

Here is updated mathjax-latex-4ht.4ht:

% mathjax-latex-4ht.sty (2020-07-27-14:09), generated from tex4ht-mathjax.tex
% Copyright 2018-2019 TeX Users Group
%
% This work may be distributed and/or modified under the
% conditions of the LaTeX Project Public License, either
% version 1.3c of this license or (at your option) any
% later version. The latest version of this license is in
%   http://www.latex-project.org/lppl.txt
% and version 1.3c or later is part of all distributions
% of LaTeX version 2005/12/01 or later.
%
% This work has the LPPL maintenance status "maintained".
%
% The Current Maintainer of this work
% is the TeX4ht Project <http://tug.org/tex4ht>.
%
% If you modify this program, changing the
% version identification would be appreciated.
\immediate\write-1{version 2020-07-27-14:09}

\RequirePackage{etoolbox,expl3,environ}

\ExplSyntaxOn
\cs_new_protected:Npn \alteqtoks #1
{
  \tl_set:Nx \l_tmpa_tl {\detokenize{#1}}
  % convert \ { to \:{  
  \regex_replace_all:nnN { \x{5C} \x{20} \x{7B} } { \x{5C} \x{3A} \x{7B} } \l_tmpa_tl
  \regex_replace_all:nnN { \x{5C} \x{5C}  \x{3A} \x{7B} } { \x{5C} \x{5C} \x{20} \x{7B} } \l_tmpa_tl
  % delete spaces before left brackets
  \regex_replace_all:nnN { \x{20} \x{7B} } { \x{7B} } \l_tmpa_tl
  % convert \\:{ back to \\ { 
  \regex_replace_all:nnN { \x{5C} \x{5C}  \x{3A} \x{7B} } { \x{5C} \x{5C} \x{20} \x{7B} } \l_tmpa_tl
  % replace < > and & with xml entities
  \regex_replace_all:nnN { \x{26} } { &amp; } \l_tmpa_tl
  \regex_replace_all:nnN { \x{3C} } { &lt; } \l_tmpa_tl
  \regex_replace_all:nnN { \x{3E} } { &gt; } \l_tmpa_tl
  \tl_set:Nx \l_tmpb_tl{ \l_tmpa_tl }
  \HCode{\l_tmpb_tl}
}
\ExplSyntaxOff

\NewConfigure{MathJaxConfig}{1}

\long\def\AltlMath#1\){\expandafter\alteqtoks{\(#1\)}\)}
\long\def\AltlDisplay#1\]{\alteqtoks{\[#1\]}\]}
\long\def\AltMathOne#1${\alteqtoks{\(#1\)}$}
% this seems a bit hacky -- we need to skip some code inserted at the
% beginning of each display math
\long\def\AltlDisplayDollars\fi#1$${\alteqtoks{\[#1\]}$$}

\newcommand\VerbMathToks[2]{%
  \alteqtoks{\begin{#2}
    #1
  \end{#2}}%
}
\newcommand\VerbMath[1]{%
\ifcsdef{#1}{%
  \RenewEnviron{#1}{%
  \NoFonts\expandafter\VerbMathToks\expandafter{\BODY}{#1}\EndNoFonts%
  }
}{}%
}
\def\fixmathjaxtoc#1{\Configure{writetoc}{\def#1{\detokenize{#1}}}}
\def\fixmathjaxsec#1{\def#1{\detokenize{#1}}}

\AtBeginDocument{%
\VerbMath{subarray}
\VerbMath{smallmatrix}
\VerbMath{matrix}
\VerbMath{pmatrix}
\VerbMath{bmatrix}
\VerbMath{Bmatrix}
\VerbMath{vmatrix}
\VerbMath{Vmatrix}
\VerbMath{cases}
\VerbMath{subequations}
\VerbMath{aligned}
\VerbMath{alignedat}
\VerbMath{gathered}
\VerbMath{gather}
\VerbMath{gather*}
\VerbMath{alignat}
\VerbMath{alignat*}
\VerbMath{xalignat}
\VerbMath{xalignat*}
\VerbMath{xxalignat}
\VerbMath{align}
\VerbMath{align*}
\VerbMath{flalign}
\VerbMath{flalign*}
\VerbMath{split}
\VerbMath{multline}
\VerbMath{multline*}
\VerbMath{equation}
\VerbMath{equation*}
\VerbMath{math}
\VerbMath{displaymath}
\VerbMath{eqnarray}
\VerbMath{eqnarray*}
\ConfigureEnv{multline}{}{}{}{}
\ConfigureEnv{multline*}{}{}{}{}
\fixmathjaxsec\left
\fixmathjaxsec\right
\fixmathjaxtoc\int
\fixmathjaxtoc\,
\fixmathjaxtoc\sin
\fixmathjaxtoc\cos
\fixmathjaxtoc\tan
\fixmathjaxtoc\arcsin
\fixmathjaxtoc\arccos
\fixmathjaxtoc\arctan
\fixmathjaxtoc\csc
\fixmathjaxtoc\sec
\fixmathjaxtoc\cot
\fixmathjaxtoc\sinh
\fixmathjaxtoc\cosh
\fixmathjaxtoc\tanh
\fixmathjaxtoc\coth
\fixmathjaxtoc\log
\fixmathjaxtoc\ln
\fixmathjaxtoc\sum

}
\endinput

The important code is this:

\cs_new_protected:Npn \alteqtoks #1
{
  \tl_set:Nx \l_tmpa_tl {\detokenize{#1}}
  % convert \ { to \:{  
  \regex_replace_all:nnN { \x{5C} \x{20} \x{7B} } { \x{5C} \x{3A} \x{7B} } \l_tmpa_tl
  \regex_replace_all:nnN { \x{5C} \x{5C}  \x{3A} \x{7B} } { \x{5C} \x{5C} \x{20} \x{7B} } \l_tmpa_tl
  % delete spaces before left brackets
  \regex_replace_all:nnN { \x{20} \x{7B} } { \x{7B} } \l_tmpa_tl
  % convert \\:{ back to \\ { 
  \regex_replace_all:nnN { \x{5C} \x{5C}  \x{3A} \x{7B} } { \x{5C} \x{5C} \x{20} \x{7B} } \l_tmpa_tl
  % replace < > and & with xml entities
  \regex_replace_all:nnN { \x{26} } { &amp; } \l_tmpa_tl
  \regex_replace_all:nnN { \x{3C} } { &lt; } \l_tmpa_tl
  \regex_replace_all:nnN { \x{3E} } { &gt; } \l_tmpa_tl
  \tl_set:Nx \l_tmpb_tl{ \l_tmpa_tl }
  \HCode{\l_tmpb_tl}
}

This is the result: