The two-dimensional (2D) kagome lattice is a versatile platform to explore exotic quantum states. The recently discovered kagome superconductors AV3Sb5 (A= Cs, Rb, K) containing vanadium kagome lattices exhibit a variety of intriguing phenomena, such as a charge density wave (CDW) with time-reversal symmetry breaking and possible unconventional superconductivity. In particular, in bulk samples the CDW displays a three-dimensional character and tends to be suppressed by external or chemical pressure. However, the characteristics and stability of the CDW order in atomically thin flakes approaching the 2D limit remain unexplored. Here, through combined electrical transport, scanning transmission electron microscopy (STEM) and Raman scattering measurements, we report an intertwined-order phase diagram of CsV3Sb5 flakes down to the monolayer. After successfully confirming the stability of the kagome lattice down to at least 4 layers, we observe a non-monotonic evolution of the CDW transition temperature TCDW with a reduction of flake thickness. TCDW first decreases to a minimum value of 72 K at 27 layers and then increases abruptly, reaching a record-high value of 120 K at 5 layers. The superconducting transition temperature (Tc) features an inverse variation with TCDW. Flakes with less than 5 layers, however, become more insulating with decreasing thickness. Raman scattering measurements reveal a weakened electron-phonon coupling with the reduction of sample thickness, suggesting that a crossover from electron-phonon coupling to dominantly electronic interactions could account for the non-monotonic thickness dependence of TCDW. Our work demonstrates the novel effects of dimension reduction and carrier doping on quantum states in thin flakes and provides crucial insights into the complex mechanism of the CDW order in the family of AV3Sb5 kagome metals.