`counterwithin` not work with custom counters

You need stepcounter in order to reset the associated counters.

documentclass{article}
usepackage[a4paper,textwidth=7in]{geometry}

geometry{textheight=6cm}% just to make a smaller picture

newcounter{lecnum}
renewcommand{thepage}{thelecnum-arabic{page}}
counterwithin{section}{lecnum}
counterwithin{equation}{lecnum}
counterwithin{figure}{lecnum}
counterwithin{table}{lecnum}

pagestyle{myheadings}

newcommand{lecture}[4]{%
  clearpage
  thispagestyle{plain}
  stepcounter{lecnum}% resets every associated counter
  setcounter{lecnum}{#1}%
  setcounter{page}{1}
  begin{center}
    framebox{%
      parbox[t]{6.28in}{%
        vspace{2mm}
        {bfseries ECCV2020hfill August 2020par}
        vspace{2mm}
        {Largehspace*{fill}Lecture #1: #2hspace*{fill}par}
        vspace{4mm}
        {itshape Lecturer: #3hfill Scribes: #4par}
        vspace{2mm}
      }%
    }
  end{center}
  markboth{Lecture #1: #2}{Lecture #1: #2}
}

begin{document}

lecture{1}{Boxes}{egreg}{Rabbittac}

This is the first lecture.

section{First}

First section with an equation
begin{equation}
a=b
end{equation}

lecture{3}{Counters}{egreg}{Rabbittac}

This is the third lecture.

section{First}

First section with an equation
begin{equation}
a=b
end{equation}

end{document}

I refactored the code for the box to use LaTeX commands, rather than plain ones. The commands bf, it and similar have been deprecated for about 25 years.