In char 0, is there a generalised version of Bertini's theorem that will ensure that for a proper map $f: Y\rightarrow X$ between smooth projective varieties and for every point $x\in X$ we can find an open set $U \ni x$ (maybe analytic) around $x$, such that a general divisor in a base point free linear system $|L|$ on $Y$ will intersect every fibre over a point in $U$ transversally. Assume in addition that $|L|$ contains horizontal divisors.
Here, I would like to think of transversal intersection with a singular fibre as that the tangent spaces at each point in the intersection add up to give tangent space of $Y$ at that point.
However, I would like it even if a general divisor $D\in |L|$ intersect only the smooth fibres close to the fibre over $x$ transversally. The fibre over $x$ may itself be singluar.