Joining two thin lines into thicker one

If you make sure the thin line width is half the thick line, you can use a to path with appropriate angles and coordinates.

\documentclass[tikz,border=20pt]{standalone}

\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
    \draw[-, line width=1mm] (0,2) to[out=30,in=180] (1,2.5cm-0.5mm) ++(0,1mm) to[out=180,in=-30] (0,3);
    \draw[-, line width=2mm]  (1,2.5) to (2,2.5);
\end{tikzpicture}
\end{document}

The following doesn't look too bad (but is in no way automated):

\documentclass[tikz,border=0pt]{standalone}

\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
    \draw[-, line width=1mm] (0,2) to (1,2.5);
    \draw[-, line width=1mm] (0,3) to (1,2.5);
    \draw[-, line width=2mm]  (0.9,2.5) to (2,2.5);
\end{tikzpicture}
\end{document}