I have a problem I have been stuck with since several weeks now, and yet I believe it should be easy to specialists.

Let $k$ be an algebraically closed field, $m$ and $n$ two integers. Let $H_1,\dots,H_m$ be $m$ (smooth if that helps) hypersurfaces of degrees $d_1, \dots,d_m$ in the projective space $\mathbb P^n$ over $k$. Let $X$ be the intersection $H_1 \cap \dots \cap H_n$, and for $\ell$ any fixed auxiliary prime different from the characteristic of $k$, let $b_i$ be the Betti numbers of $X$ (rank of $H^i_\text{ét}(X,\mathbb Z_\ell)$). The question is:

Is there a constant $c(m,d_1,\dots,d_n)$ depending on the number of hypersurfaces $m$ and their degrees $d_i$ but not on the dimension $n$ of the ambient projective space such that all the Betti numbers $b_i$ are less than $c(m,d_1,\dots,d_n)$?

I am actually interested in the case where $b_1=\dots=b_m=2$, that is $X$ is an intersection of $m$ quadrics. For example, when $m=1$ and $d_1=2$, $X$ is a smooth quadrics, the cohomology of $X$ is well-known (see e.g. Miles Reid's thesis) and the Betti numbers are always $0$, $1$ or $2$, independently of the dimension $n-1$ of $X$. Still when $m=1$ but with $d_1$ arbitrary, I know that by Lefschetz's theorem $b_i=0$ or $1$ except for $i=n-1$, and I presume there is a formula for $b_{n-1}$ in terms of the degree $d_1$, but independent of the dimension $n$ -- Yet i couldn't find this formula in my memory nor on the web.

Though I am primarily interested to a general answer (in the case of quadrics), please feel free to assume anything that can help you ($X$ smooth, complete intersection, $m=1$, $m=2$ etc.) or replace the étale cohomology by your preferred one.

PS: after some hesitation, I have added the number theory tag because when the $H_i$ are defined over a finite field $\mathbb F_q$ (and $k$ is the algebraic closure of $\mathbb F_q$) the question is closely related (by Grothendieck's Lefschetz fixed point formula) to bounds on the number of points on $X(\mathbb F)$. For example, a positive answer would imply that for $n$ large enough with respects to $m$ and the $d_i$, $X(\mathbb F_q)$ is not-empty.