Polymers that extend covalently in two dimensions have attracted recent attention as a means of combining the mechanical strength and in-plane energy conduction of conventional two-dimensional (2D) materials with the low densities, synthetic processability, and organic composition of their one-dimensional counterparts. Efforts to date have proven successful in forms that do not allow full realization of these properties, such as polymerization at flat interfaces or fixation of monomers in immobilized lattices. A frequently employed synthetic approach is to introduce microscopic reversibility, at the cost of bond stability, to achieve 2D crystals after extensive error correction. Herein we demonstrate a synthetic route to 2D irreversible polycondensation directly in the solution phase, resulting in a covalently bonded 2D polymer that is chemically stable and highly processable. Further fabrication offers highly oriented, free-standing films which exhibit exceptional 2D elastic modulus and yield strength at 12.7 ± 3.8 GPa and 488 ± 57 MPa, respectively. The highly planar asymmetry of 2D molecules is evidenced by chemical atomic force microscopy, and molecular alignment in nanofilms is supported by a polarized photoluminescence centered at 580 and 680 nm from different dipole transitions. We have also made substantial progresses in the theory of 2D chain growth. We performed a chemical kinetic simulation study to understand 2D polymerization in homogeneous solution with irreversible chemical steps. We show that reaction sites for polymerization in 2D always scale unfavourably compared to 3D, growing as molecular weight to the 1/2 power versus 2/3 power for 3D. However, certain mechanisms can effectively suppress out-of-plane defect formation and subsequent 3D growth. We consider two such mechanisms, which we call bond-planarity and templated autocatalysis. In the first, although single bonds can easily rotate out-of-plane to render polymerization in 3D, some double-bond linkages prefer a planar configuration. In the second mechanism, stacked 2D plates may act as van der Waals templates for each other to enhance growth, which leads to an autocatalysis. When linkage reactions possess a 1000:1 selectivity (γ) for staying in plane versus rotating, solution-synthesized 2D polymers can have comparable size and yield with those synthesized from confined polymerization on a surface. Autocatalysis could achieve similar effects when self-templating accelerates 2D growth by a factor β of 106. A combined strategy relaxes the requirement of both mechanisms by over one order of magnitude. We map the dependence of molecular weight and yield for 2D polymer on the reaction parameters, allowing experimental results to be used to estimate β and γ. Our calculations show for the first time from theory, the feasibility of producing two-dimensional polymers from irreversible polymerization in solution. These new synthetic routes, now experimentally validated, provides exciting, new opportunities for 2D polymers in applications ranging from composite structures to barrier coating materials.