How can I get sharp corners inside and rounded corners outside a tcolorbox?

You can replace the frame code so that only the left border is drawn, which removes the artefacts and allows you to have the sharp internal corners:

\documentclass{article}

\usepackage[skins]{tcolorbox}

\newtcolorbox{blueleftbox}{%
  enhanced,
  boxrule=0pt,
  leftrule=5pt,
  sharp corners=west,
  frame code={
    \fill[blue]
      ([xshift=5pt]frame.north west) --
      ([xshift=5pt]frame.south west) --
      ([xshift=1mm]frame.south west) arc
      [radius=1mm,start angle=-90,end angle=-180] --
      ([yshift=-1mm]frame.north west) arc
      [radius=1mm,start angle=180,end angle=90]--
      cycle;
  }
}

\begin{document}

\begin{blueleftbox}
  This is a tcolorbox!
\end{blueleftbox}

\end{document}