  In this article, we propose a fully-discrete scheme for the numerical
solution of a nonlinear time-fractional biharmonic problem. This problem is
first converted into an equivalent system by introducing a new variable. Then
spatial and temporal discretizations are done by the weighted $b$-spline method
and $L2$-$1_\sigma$ approximation, respectively. The weighted $b$-spline method
uses weighted $b$-splines on a tensor product grid as basis functions for the
finite element space and by construction, it is a mesh-free method. This method
combines the computational benefits of $b$-splines and standard mesh-based
elements. We derive $\alpha$-robust \emph{a priori} bound and convergence
estimate in the $L^2(\Omega)$ norm for the proposed scheme. Finally, we carry
out few numerical experiments to support our theoretical findings.
