Complex turbulent motions of magnetized gas are ubiquitous in the interstellar medium. The source of this turbulence, however, is still poorly understood. Previous work suggests that compression caused by supernova shockwaves, gravity, or cloud collisions, may drive the turbulence to some extent. In this work, we present threedimensional (3D) magnetohydrodynamic (MHD) simulations of contraction in turbulent, magnetized clouds from the warm neutral medium (WNM) of the ISM to the formation of cold dense molecular clouds, including radiative heating and cooling. We test different contraction rates and find that observed molecular cloud properties, such as the temperature, density, Mach number, and magnetic field strength, and their respective scaling relations, are best reproduced when the contraction rate equals the turbulent turnover rate. In contrast, if the contraction rate is significantly larger (smaller) than the turnover rate, the compression drives too much (too little) turbulence, producing unrealistic cloud properties. The relation σ 2 s = ln(1 + b 2 M 2 ) between logarithmic density fluctuations (σ s ) and turbulent Mach number (M) is found to be consistent with previous theoretical models that were based on artificially-driven isothermal turbulence. Here we find that the effective turbulence driving parameter of contraction-driven MHD turbulence subject to heating and cooling grows from solenoidal (b ∼ 1/3) to compressive (b ∼ 1) during the contraction. Overall, the physical properties of the simulated clouds that contract at a rate equal to the turbulent turnover rate, indicate that large-scale contraction induced by processes such as supernova shockwaves, gravity, spiral-arm compression, or cloud collisions, may explain the origin and evolution of turbulence in the ISM.