For linear inverse problems $Y=\mathsf{A}μ+ξ$, it is classical to recover the unknown signal $μ$ by iterative regularisation methods $(\widehat μ^{(m)}, m=0,1,\ldots)$ and halt at a data-dependent iteration $τ$ using some stopping rule, typically based on a discrepancy principle, so that the weak (or prediction) squared-error $\|\mathsf{A}(\widehat μ^{(τ)}-μ)\|^2$ is controlled. In the context of statistical estimation with stochastic noise $ξ$, we study oracle adaptation (that is, compared to the best possible stopping iteration) in strong squared-error $E[\|\hat μ^{(τ)}-μ\|^2]$. For a residual-based stopping rule oracle adaptation bounds are established for general spectral regularisation methods. The proofs use bias and variance transfer techniques from weak prediction error to strong $L^2$-error, as well as convexity arguments and concentration bounds for the stochastic part. Adaptive early stopping for the Landweber method is studied in further detail and illustrated numerically.