Recent developments in storage -- especially in the area of resistive random access memory (ReRAM) -- are attempting to scale the storage density by regarding the information data as two-dimensional (2D), instead of one-dimensional (1D). Correspondingly, new types of 2D constraints are introduced into the input information data to improve the system reliability. While 1D constraints have been extensively investigated in the literature, the study for 2D constraints is much less profound. Particularly, given a constraint $\mathcal{F}$ and a design of 1D codes whose codewords satisfy $\mathcal{F}$, the problem of constructing efficient 2D codes, such that every row and every column in every codeword satisfy $\mathcal{F}$, has been a challenge. This work provides an efficient solution to the challenging coding problem above for the binary bounded-weight constrained codes that restrict the maximum number of $1$'s (called {\em weight}). Formally, we propose a universal framework to design 2D codes that guarantee the weight of every row and every column of length $n$ to be at most $f(n)$ for any given function $f(n)$. We show that if there exists a design of capacity-approaching 1D codes, then our method also provides capacity-approaching 2D codes for all $f=ω(\log n)$.