How to draw a tikz smooth path connecting points on an image?

Here is an example if using the hobby package that can draw smooth curve through some given points.

I defined some intermediate points (A'), (A'') and (D'), (D'') that help to shape near the two points that are connected via a straight line. You can adjust those points to obtain the desired shape.

Code:

\documentclass[border=2pt]{standalone}

\usepackage{tikz}
\usetikzlibrary{hobby}
 
\begin{document} 
\begin{tikzpicture} 
%\node[anchor=south west,inner sep=0] 
%at (-3,-2) {\includegraphics[width=\textwidth]{euler.png}};  

\coordinate (A) at (1.2,1.2);
\coordinate (A')  at ([shift={(-0.5,  0.5)}]A);
\coordinate (A'') at ([shift={(-0.5, -0.5)}]A);

\coordinate (B) at (1,-1.9);
\coordinate (C) at (1.5,4);

\coordinate (D) at (6.2,.9);
\coordinate (D')  at ([shift={(-0.5,  1.0)}]D);
\coordinate (D'') at ([shift={(-0.5, -1.0)}]D);

\draw [fill=blue] (A) circle (3pt) node [above right] {A};
\draw [fill=blue] (B) circle (3pt) node [below left]  {B};
\draw [fill=blue] (C) circle (3pt) node [above left]  {C};
\draw [fill=blue] (D) circle (3pt) node [right]       {D}; 

\draw [red, thick] (D) -- (A)
    to [curve through={(A') (C) (D')}]
    (D);
    
\draw [red, thick] (D) -- (A)
    to [curve through={(A'') (B) (D'')}]
    (D);
\end{tikzpicture}
\end{document} 

I suggest giving the bridges coordinate names. Then you can use the command

\draw (x) to[out=<angle>,in=<angle>] (y);

so that the paths go over the bridges.

I drew red circles to indicate the bridge coordinates, but you can of course remove them by deleting the \fill[red]... commands. I also gave coordinate names to your blue circles to make adjustments easier.

Here is the code:

\documentclass[tikz]{standalone} 
\begin{document} 
\begin{tikzpicture} 
\node[anchor=south west,inner sep=0] 
at (-3,-2) {\includegraphics[width=\textwidth]{mOizq.png}};  
\coordinate (a) at (-.25,-.35);
\coordinate (b) at (2.6,-.92);
\coordinate (c) at (.32,2.82);
\coordinate (d) at (3.05,2.75);
\coordinate (e) at (4.3,.88);
\coordinate (f) at (5.3,-.27);
\coordinate (g) at (5.7,2.1);
\coordinate (A) at (1.2,1.2);
\coordinate (B) at (1,-1.9);
\coordinate (C) at (1.5,4);
\coordinate (D) at (6.2,.9);
\fill[red] (a) circle(2pt);
\fill[red] (b) circle(2pt);
\fill[red] (c) circle(2pt);
\fill[red] (d) circle(2pt);
\fill[red] (e) circle(2pt);
\fill[red] (f) circle(2pt);
\fill[red] (g) circle(2pt);
\draw [fill=blue] (A) circle (3pt);
\draw [fill=blue] (B) circle (3pt);
\draw [fill=blue] (C) circle (3pt);
\draw [fill=blue] (D) circle (3pt);
\begin{scope}[line width=1, blue];
\draw (A) to[out=-135,in=60] (a) to[out=-120,in=180] (B) to[out=0,in=-90] (b) to[out=90,in=180] (e) to[out=0,in=180] (D) to[out=-135,in=75] (f) to[out=-115,in=0] (B);
\draw (A) to[out=135,in=-115] (c) to[out=75,in=180] (C) to[out=0,in=60] (d) to[out=-120,in=180] (e) to[out=180,in=0] (A);
\draw (C) to[out=0,in=115] (g) to[out=-75,in=135] (D);
\end{scope}
\end{tikzpicture}
\end{document}