It is more related to the needed strength of the magnets to keep the particles on track and also the synchrotron radiation. The first issue is that the lower the radius of the circle, the stronger magnetic field is needed because of the increased centrifugal forces. The second issue is that synchrotron radiation leads to loss of energy. It is again proportional to the acceleration perpendicular to the vector of the velocity of the particle [0].

[0] https://en.wikipedia.org/wiki/Synchrotron_radiation

The particles are steered round the accelerator with magnetic fields from superconducting magnets.

Higher speeds (and thus energies) can be achieved by increasing the circular circuit radius by limiting the force that has to be applied and the energy loss in the acceleration (change of direction around the circle).

The particles that go around radiate their energy away, which is a function of the radius of the circle. It is called synchrotron radiation and is the main limitation factor of these kind of accelerators.

Turning a particle moving at half light speed requires some pretty big magnets; the bigger the faster and tighter the turn.

The bigger turning circle lets them go faster with the same magnets.