We think the following is true:

For all sufficiently large primes $p$ and all natural $g \ge 1$, there exists affine plane curve $f(x,y)=0$ over $\mathbb{F}_p$ which is non-singular, absolutely irreducible, of genus $g$ and it doesn't have any rational points.

Is it true?

Is it known?

This doesn't violate Hasse-Weil bound, because the bound
requires **SMOOTH** projective model and our examples have very
few singular points on the projective model.

For $p=13,g=1$ check this question and comments

**Added** Example of pointless non-sigular affine curves for $g=1$ defined
by two equations.
Let $f_0=x^3+y^2-1,f_1=z(x^p-x)-1$ and the curve $C : f_0=0,f_1=0$.
Then $f_1$ is linear in $z$ so $C$ is birationally equivalent to $f_0=0$.

The curve is pointless because $x^p-x$ is zero modulo $p$.

To get a single equation for the curve set $f$ the resultant of $f_0$ and $f_1$ wrt $x$, experimentally it is irreducible.

affineplane curves. If this is the case you should write it in big bold capital letters in the question, because everyone (and certainly the three people who reacted so far) will interpret “curve” as “projective curve”! $\endgroup$ – Gro-Tsen May 29 '19 at 12:29