We map out the QCD crossover line $\frac{T_c(\mu_B)}{T_c(0)} = 1 - \kappa_2 \left(\frac{\mu_B}{T_c(0)} \right)^2 - \kappa_4 \left( \frac{\mu_B}{T_c(0)} \right)^4 + \mathcal{O}(\mu_B^6)$ for the first time up to $\mathcal{O}(\mu_B^4)$ for a strangeness neutral system by performing a Taylor expansion of chiral observables in temperature $T$ and chemical potentials $\mu$. At vanishing chemical potential, we report a crossover temperature $T_c(0) = (156.5 \pm 1.5)\;\mathrm{MeV}$ defined by the average of several chiral susceptibilities. For a system with thermal conditions appropriate for a heavy-ion collision, we determined a curvature from the subtracted condensate as $\kappa_2 = 0.0120(20)$ and from the disconnected susceptibility as $\kappa_2 = 0.0123(30)$. The next order $\kappa_4$ is significantly smaller. We also report the crossover temperature as a function of the chemical potentials for: baryon-number, electric charge, strangeness and isospin. Additionally, we find that $T_c(\mu_B)$ is in agreement with lines of constant energy density and constant entropy density. Along this crossover line, we study net baryon-number fluctuations and show that their increase is substantially smaller compared to that obtained in HRG model calculations. Similarly, we analyze chiral susceptibility fluctuations along the crossover line and show that these are constant. We conclude that no signs for a narrowing of the crossover region can be found for baryon chemical potential $\mu_B < 250\;\mathrm{MeV}$.