Well, why not just the following, then?
\parindent=0pt
\newdimen\linewidth
\linewidth=4in
\newdimen\colsep
\colsep=1em
\newdimen\colwidth
\colwidth=0.5\linewidth
\advance\colwidth by -\colsep
\def\singlebox#1{%
\vtop{%
\hsize=\colwidth
\hangindent=1em \hangafter=1
\leavevmode #1\par
\kern -\prevdepth
}%
}
\def\twocols#1#2{%
\par
\hbox to\linewidth
{%
\singlebox{#1}\hfill\singlebox{#2}%
}%
\prevdepth = 0pt
}
\twocols
{This is a very long line that will run over over over over over over jjjjj over}
{What is wrong with the spacing?}
\twocols
{What is wrong with the spacing?}
{This is a very long line that will run over}
More text.
% {\tracingall\showlists}
\bye
Quite obviously, since it “kills” the depth of the bottom line inside the \twocols construction, this solution is not adequate when the text contains unusual descenders (as egreg has immediately noticed). This answer was intended just as a streamlining of jfbu’s idea.
A possible refinement is:
\parindent=0pt
\newdimen\linewidth
\linewidth=4in
\newdimen\colsep
\colsep=1em
\newdimen\colwidth
\colwidth=0.5\linewidth
\advance\colwidth by -\colsep
\def\singlebox#1{%
\vtop{%
\hsize=\colwidth
\hangindent=1em \hangafter=1
\leavevmode #1\par
\kern -%
\ifdim\prevdepth<\maxdepth
\prevdepth
\else
\maxdepth
\fi
}%
}
\def\twocols#1#2{%
\par
\hbox to\linewidth
{%
\singlebox{#1}\hfill\singlebox{#2}%
}%
\prevdepth = 0pt
}
\twocols
{This is a very long line that will run over over over over over over jjjjj over}
{What is wrong with the spacing?}
\twocols
{What is wrong with the spacing?}
{This is a very long line that will run over}
More text.
\twocols
{This is a very long line that will run over over over over over over jjjjj
over $\Bigg|$}
{What is wrong with the spacing?}
\twocols
{What is wrong with the spacing?}
{This is a very long line that will run over}
Still more text.
% {\tracingall\showlists}
\bye
The output produced by this amended code is

Please note that this does work at the bottom of pages too. Moreover, since the OP didn’t insert the \parskip glue above the \twocols construction, I did not too; if desired, an explicit \vskip\parskip can be added just before the \hbox command in the definition of \twocols.