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Let $Y\subseteq X$ be a properly embedded hypersurface. Choose local defining smooth functions $f_{\alpha }$ with nonzero differentials along $Y$. On overlaps they differ by nowhere-zero smooth factors. These transition factors define a real line bundle $L\to X$ and a transverse section $s$ whose zero set is $Y$.

The derivative of $s$ along its zero set is intrinsic and gives a surjective bundle mapIts kernel is $TY$, so it induces the vector bundle isomorphismThus the restriction of the defining line bundle is the normal bundle. These bundles have different original base spaces: $L$ lives over $X$, while $N_{Y/X}$ lives over $Y$. A global real-valued regular defining function would trivialize that restriction, explaining the obstruction from a nontrivial normal bundle.