A bounded linear operator $T$ on a Banach space $X$ is called subspace-supercyclic for a closed and non-zero subspace $M$ of $X$ if there exists a vector $x\in X$ whose projective orbit $\mathbb{C}{orb}(T, x)\cap M= \{\beta T^n(x):~~\beta \in \mathbb{C},n=0, 1, 2, \ldots\}\cap M$ is dense in $M$. Let $1\le p<\infty$ and $\mathcal{B}$ is a Borel $\sigma$-algebra. For a $\sigma$-subalgebra $\mathcal{A}$ of $\mathcal{B}$, the conditional expectation with $\mathcal{A}$ is the mapping $f\to E^{\mathcal{A}}f$. Also, for a second countable locally compact group $G$ and $g\in G$, $\delta_g$ is the unit point mass measure at $g$. A bounded Borel measurable function $v:G\to (0, \infty)$ is called {\em weight}. In this paper, we characterize the subspace-supercyclicity of the conditional weighted translation $R_{g, v}:L^p(\mathcal{B})\to L^p(\mathcal{A})$ defined by $R_{g, v}(f):=E^{\mathcal{A}}(v f*\delta_g)$. % where the convolution %$$(f*\delta_g)(x)=\int_G f(xy^{-1})d\delta_g(y)=f(xg^{-1})\qquad x\in G.$$ First, we prove that if $g\in G$ is a torsion element, then $R_{g, v}$ cannot be subspace-supercyclic with respect to $L^p(\mathcal{A})$. Besides, %in the case that $g\in G$ is an %aperiodic element, we reveals that $R_{g, v}$ is subspace-supercyclic if $g$ is an aperiodic element. For $\alpha\in \mathbb{C}\setminus \{0\}$, the %The concepts of subspace-hypercyclicity and subspace-mixing are also studied for $\alpha R_{g, v}.$ %with $\alpha\in \mathbb{C}\setminus \{0\}$. Also, we specify the Ces$\grave{a}$ro subspace-hypercyclicity for $R_{g, v}$ and offer conditions for the adjoint of $R_{g, v}$ to be subspace-supercyclic with respect to $L^2(\mathcal{A}g^{-1})$.
