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This paper studies the existence and uniqueness of solutions for an irregular fractional boundary value problem (IFBVP) of order $q\in(3/2,2]$. In order to apply Rus's fixed point theorem with the maximum and $L^2$metrics, we derive new estimates for the associated Green's function. We evaluate the squared Green’s function in a double integral to obtain the constant in the $L^2$estimate. Evaluating the mixed terms of this integral makes the appearance of the hypergeometric function 3F2(1), in which we apply classical transformation formulas to obtain finite sum evaluations of these terms, yielding the explicit constant $\mathcal{K}_2(q,\rho,\theta)$. Using these estimates, we establish existence and uniqueness of solutions to the IFBVP under a Lipschitz condition on the nonlinear function $f$with constant $L>0$, provided that $L\sqrt{\mathcal{K}_2(q,\rho,\theta)}<1$. Numerical comparisons with an existing condition involving $\Lambda(q,\rho,\theta)$show that our condition is less restrictive for some parameter values, while the existing condition is less restrictive for others.