In this paper, the authors study the existence and regularity of normalized solutions to the doubly nonlocal equation $$ -\Delta u=\lambda u+\mu_1\left(I_\alpha *|u|^p\right)|u|^{p-2} u+\mu_2\left(I_\beta *|u|^p\right)|u|^{p-2} u \text { in } \Bbb{R}^3, \tag1 $$ with a prescribed mass $\int_{\Bbb{R}^3} u^2=a>0$. Here $\lambda \in \Bbb{R}$ arises as a Lagrange multiplier, $\alpha, \beta \in(0,3)$ with $\alpha \leqslant \beta$, $ p \in[2_*^\beta, 2_\alpha^*]$, $I_\alpha$ and $I_\beta$ are Riesz potentials, and $\mu_1, \mu_2>0$ are constants. In particular, the authors estimate the energy level ingeniously and consider the existence of normalized solutions to the above equation with Hardy-Littlewood-Sobolev lower critical exponent $p=2_*^\beta$ or upper critical exponent $p=$ $2_\alpha^*$. \par Problem (1) comes from studying the standing waves of $$ i \partial_t \Psi+\Delta \Psi+\left(V *|\Psi|^p\right)|\Psi|^{p-2} \Psi=0 \text { in } \Bbb{R}_{+} \times \Bbb{R}^3, \tag2 $$ where $t$ denotes the time, $\Psi\: \Bbb{R}_{+} \times \Bbb{R}^3 \rightarrow \Bbb{C}$ is a complex-valued function, $\Bbb{R}_{+}\coloneq [0, \infty)$, and $V=\mu_1 I_\alpha+\mu_2 I_\beta$. Then $\Psi(t, x)=e^{-i \lambda t} u(x)$ solves (2) if and only if $u$ is a solution of (1). If $\alpha=\beta=p=2$, then equation (1) is called the Choquard equation and it describes the quantum mechanics of a polaron at rest [S.~I. Pekar, {\it Untersuchungen über die Elektronentheorie der Kristalle}, Akademie-Verlag, Berlin, 1954, \doi{10.1515/9783112649305}], and an electron trapped in its own hole, as an approximation to the Hartree-Fock theory of a one-component plasma [E.~H. Lieb, Studies in Appl. Math. {\bf 57} (1976/77), no.~2, 93--105; MR0471785]. \par The main result of this article is: \par Theorem 1. Let $\mu_i>0$, $i=1,2$ and $0<\alpha<\beta<\min \{\alpha+2,3\}$. Then, for $p \in(2_*^\beta, \overline{\alpha})$ or $p \in(\overline{\beta}, 2_\alpha^*)$, the equation (1) has a radial ground state normalized solution $u_a$ with $\lambda_a<0$. If $p=\overline{\alpha}$ and $$ \mu_1 C_{H G}(\alpha) a^{\overline{\alpha}-1}<\overline{\alpha} $$ or if $p \in(\overline{\alpha}, \overline{\beta})$ and $$ \mu_2 C(\beta, p) a^{\frac{(\beta-\alpha)(p-1)}{2\left(\gamma_\alpha p-1\right)}}<2 p\left(\frac{\mu_1 C(\alpha, p)(\beta-\alpha)}{2 p\left(1-\gamma_\beta p\right)}\right)^{\left(\gamma_\beta p-1\right) /\left(\gamma_\alpha p-1\right)} \frac{\gamma_\alpha p-1}{\beta-\alpha} $$ or if $p=\overline{\beta}$ and $$ \mu_2 C_{H G}(\beta) a^{\overline{\beta}-1}<\overline{\beta}, $$ then the conclusion is still true, and $C(\xi, p)=C_{H G}(\xi)$ when $p=\overline{\xi}$, $\xi=\alpha, \beta$. \par If $p=2_*^\beta$, then there exists $\mu_a>0$ such that the equation (1) has a radial ground state normalized solution $u_a$ with $\lambda_a<0$ when $\mu_1 \geqslant \mu_a$. \par If $p=2_\alpha^*$ and $(3+\alpha) \beta<6 \alpha$, then there exists $\mu_a^{\prime}>0$ such that the equation (1) has a radial ground state normalized solution $u_a$ with $\lambda_a<0$ when $\mu_2 \geqslant \mu_a^{\prime}$. \par Moreover, the normalized solution $u_a$ of the equation (1) with $p \in(2_*^\beta, 2_\alpha^*]$ and $\lambda_a<0$ belongs to $W_{\roman{loc}}^{2, \sigma}(\Bbb{R}^3)$ for all $\sigma \in[1, \infty)$.