In this study we address whether the age--metallicity relation (AMR) deviates from the expected trend of metallicity increasing smoothly with age. We also show the presence (or absence) of two populations, as recently claimed using a relatively small dataset. Moreover, we studied the Milky Way thin disk's chemical evolution using solar twins, including the effect of radial migration and accretion events. In particular, we exploited high-resolution spectroscopy of a large sample of solar twins in tandem with an accurate age determination to investigate the Milky Way thin disk age--metallicity relationship. Additionally, we derived the stars' birth radius and studied the chemical evolution of the thin disk. We discovered that statistical and selection biases can lead to a misinterpretation of the observational data. An accurate accounting of all the uncertainties led us to detect no separation in the AMR into different populations for solar twins around the Sun ($-0.3 < Fe/H < 0.3$ dex). This lead us to the conclusion that the thin disk was formed relatively smoothly. For the main scenario of the Milky Way thin disk formation, we suggest that the main mechanism for reaching today's chemical composition around the Sun is radial migration with the possible contribution of well-known accretion events such as Gaia -Enceladus/Sausage (GES) and Sagittarius (Sgr).