Not every force is manifested through the curvature of spacetime, only gravity. Take the electromagnetic field, for example. It has energy-momentum density, so it affects the gravitational field and hence the movement of particles. But this is not the only effect it has: on a charged particle, it also acts directly through the electromagnetic force.
Mathematically, for a particle of mass $m$ and charge $q$ in an EM field $F_{\mu\nu}$, the geodesic equation is modified to
$$\frac{d u^\mu}{d\tau} + \Gamma^\mu{}_{\nu\lambda} u^\nu u^\lambda = \frac{q}{m} F^\mu{}_\nu u^\nu.$$
If there was no electromagnetic field, the right hand side would be zero, and you would just have the geodesic equation with the gravitational field $\Gamma^\mu{}_{\nu\lambda}$. If there is an EM field, then two things happen. For one, its energy-momentum influences the gravitational field, so that $\Gamma^\mu{}_{\nu\lambda}$ changes. But it also directly applies a force on the particle, given by the right hand side of the equation, and this effect is usually much bigger. The particle doesn't follow a geodesic anymore.
