In the human glucose-insulin regulatory system, diverse metabolic issues can arise, including diabetes type I and type II, hyperinsulinemia, hypoglycemia, etc. Therefore, the analysis and characterization of such a biological system is a must. It is well known that mathematical models are an excellent option to study and predict natural phenomena to some extent. In this way, fractional-order theory provides generalizations for derivatives and integrals to arbitrary orders giving us a framework to add memory properties and an additional dimension to the mathematical models to approximate real-world phenomena with higher accuracy.In this work, we study the glucose and insulin governing mechanisms using a fractional-order version of a mathematical model. Applying the fractional-order Caputo derivative, we can investigate different concentration rates among insulin, glucose, and healthy beta cells. Additionally, the model incorporates two time-lags to represent the elapsed time of two processes, i.e., the delay in secrete insulin for a blood glucose increment and the lag to get a glucose reduction caused by raised insulin level. Analytical results of the equilibrium points and their corresponding stability are given. Numerical results, including phase portraits and bifurcation diagrams, reveal that the fractional-order increases the chaotic regions, leading to potential metabolic problems. Vice versa, the system seems to work correctly when the behavior evolves to periodic windows.