Are composite materials, which are being widely used in newer aircraft, better at handling bird strikes?
Out of the box, no. They are worse at handling birdstrikes. But with some thought they can be made much better.

All materials have an elastic range of deformation, and if that is not exceeded, they will assume their original shape without damage. Composites fail if this range is exceeded. First the fibers with the highest elongation fail, which enables the other fibers to take more load, but at some point too few fibers remain, and the part fails catastrophically. This is audible: The first small failures cause a faint crackling, and the eventual failure a loud bang.

Metals add a plastic deformation before ultimate failure, and this enables them to absorb a lot of energy in the process. Since energy is force times distance, the small elastic deformation will not absorb much energy. However, the plastic deformation is magnitudes bigger and makes metals the much better energy absorbers. Even a deformed structure can take a sizable fraction of the load it took before deformation. A birdstrike will cause a dent, but the rest of the structure will not see much of the load since the dent-forming could absorb a lot of energy. However, the changed contour can provoke changed aerodynamic loads, so new failure modes might result.

Impact loads on composites have a second disadvantage: The impact creates shockwaves which travel through the structure and cause delamination (separation between fibers and resin matrix) when they hit a boundary. Therefore, the biggest damage occurs on the opposite side of the impact. A birdstrike might not leave a visible dent on the outside, but can shatter the material on the opposite side such that the structure will break at a fraction of the load it would have carried without the bird strike.

Therefore, some composite wings use metallic leading edges to protect the composite structure from birdstrikes. But also the composite structure itself can be improved: If very tough fibers like Kevlar or Spectra are added, the impact energy will not break those fibers, and the load will be distributed over much more of the resin matrix. This will delay delamination, and when it happens, it will be spread over a bigger area and cause the fibers to break at multiple locations. The energy needed for the widespread delamination and the multiple fiber failures will increase the energy absorption dramatically. This structure must be designed as a protective armor, so the load-carrying structure below is not affected. Proper crashworthy design with composites allows to make the structure even tougher than a comparable metallic structure.