Embedded resolution of curves on smooth varieties

As far as I understand, embedded resolution of singularities means the following: given a variety $X$ over an algebraically closed field, and a closed subvariety $Y$, there exists a birational map $f:X'\to X$ such that $X'$ and the proper transform $Y'$ of $Y$ are smooth. Sometimes a bit more is required (e.g. $Y'$ transverse to the exceptional locus), so I'm a bit confused about the terminology.

We know that we can do this if either (1) the base field is of characteristic zero, or (2) $X$ is a surface and $Y$ is a curve, and maybe in some other low-dimensional situations.

Question. Can we always find an embedded resolution when $X$ is smooth and $Y$ is a curve?

This should be fine at least for the weaker notion you mention. One can always resolve the singularities of a curve by blowing up a sequence of closed points (on the curve). Hence if you embed the curve in a regular ambient space $X$, you can always do the following. Choose the point on $Y \subseteq X$ you were going to blowup, blowup that closed point on $X$, and notice that the resulting $X_1$ is still regular (since you blew up a regular center). Repeat to get $\ldots \to X_3 \to X_2 \to X_1 \to X$.
However, you could ask for more as you said! In particular that the exceptional divisor of $X' \to X$ is SNC and meets $Y'$ also transversely. I don't know if this is true in the generality you want.