Critical behavior and evidence of dimensional crossover in quasi-two-dimensional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Li</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:msub> <mml:mi>FeSiO</mml:mi> <mml

We report thermal expansion and heat capacity studies on Li$_2$FeSiO$_4$ single crystals which enable us to investigate the critical behavior in the magnetically quasi-two-dimensional (2D) material. Pronounced $λ$-shaped anomalies at the magnetic ordering temperature $T_{\rm N}$ imply significant magneto-elastic coupling. Our analysis of both the thermal expansion and the specific heat data implies the crossover from 2D Ising-like behavior for $|(T-T_{\rm N})/T_{\rm N}|>0.3$ to 3D Ising behavior \rev{below $\simeq 1.3\times T_{\rm N}$. The 2D-like behavior is further supported by density functional calculations which show minimal dispersion perpendicular to the crystallographic $ac$ planes of the layered structure, thereby indicating the 2D nature of magnetism at higher temperatures.} Our results extend the available model materials of quasi-2D magnetism to a high-spin $S=2$ system with tetrahedrally coordinated Fe$^{2+}$-ions, thereby illustrating how magnetic order evolves in a 2D Ising-like system with orbital degrees of freedom.