We construct models of the structure and evolution of the Sun which include variable magnetic fields and turbulence. The magnetic effects are (1) magnetic pressure, (2) magnetic energy, and (3) magnetic modulation to turbulence. The effects of turbulence are (1) turbulent pressure, (2) turbulent kinetic energy, and (3) turbulent inhibition of the radiative energy loss of a convective eddy, and (4) turbulent generation of magnetic fields. Using these ingredients we construct five types of solar variability models (including the standard solar model) with magnetic effects. These models are in part based on three-dimensional numerical simulations of the superadiabatic layers near the surface of the Sun. The models are tested with several sets of observational data, namely, the changes of (1) the total solar irradiance, (2) the photospheric temperature, (3) radius, (4) the position of the convection zone base, and (5) low- and medium-degree solar oscillation frequencies. We find that turbulence plays a major role in solar variability, and only a model that includes a magnetically modulated turbulent mechanism can agree with all the current available observational data. We find that because of the somewhat poor quality of all observations (other than the helioseismological ones), we need all data sets in order to restrict the range of models.