Starting a proof with \paragraph

Add an \item or ~ before \paragraph. For example:

\begin{document}
\begin{proof}
\item 
\paragraph{This bit in bold.} Now this part of the proof.
\end{proof}
\end{document}

I think the best way is to define a new environment, let's say breakproof, that acts like proof but starts with a new line after the word "Proof".

So you can write

\documentclass{article}
\usepackage{amsthm}

\makeatletter
\newenvironment{breakproof}[1][\proofname]{\par
  \pushQED{\qed}%
  \normalfont \topsep6\p@\@plus6\p@\relax
  \trivlist
  \item[\hskip\labelsep
        \itshape
    #1\@addpunct{.}]\ignorespaces\item
}{%
  \popQED\endtrivlist\@endpefalse
}
\makeatother

\begin{document}
  \begin{breakproof}
    \textbf{This bit in bold.}
    \newline
    Now this part of the proof.
  \end{breakproof}
\end{document} 

and the result is:

Otherwise, you can redefine the proof environment (bad practice, in my opinion) and use it as before (the result is the same):

\documentclass{article}
\usepackage{amsthm}

\makeatletter
\renewenvironment{proof}[1][\proofname]{\par
  \pushQED{\qed}%
  \normalfont \topsep6\p@\@plus6\p@\relax
  \trivlist
  \item[\hskip\labelsep
        \itshape
    #1\@addpunct{.}]\ignorespaces\item
}{%
  \popQED\endtrivlist\@endpefalse
}
\makeatother

\begin{document}
  \begin{proof}
    \textbf{This bit in bold.}
    \newline
    Now this part of the proof.
  \end{proof}
\end{document}