Agreed with kkardster's earlier opinion, posted sample having full exif information would be crucial for members to diagnose your issue and suggest any recommandation.
However, if you choose to use small size image (assume you did not intentionally scale it down) or under high compression, you might never get great image.
For the sort of posted samples, they are either BIF or fast moving sport. Both require fast shutter speed to freeze their action/movement. for the BIF eagle shot, 1/500" is definitely far too slow.
To capture a well lit image, it must be properly exposed. It is determined by a combined effect of shutter speed, f/stop and ISO.
When we need a fast shutter speed for your sort of shooting, under the same lighting condition we must either use a larger f/stop (like f/1.4, f/2.8 etc instead of f/5.6, f/8 etc and obviously is a matter of hardware limitation) and/or higher ISO (that would result noisier image affecting the sharpness of an image) to compensate for the reduced amount of light reaches the sensor because of a shorter exposure time.
TZ80 uses a 1/2.3" sensor, which is classified among the small ones and is not the best on high ISO performance:

Extracted from Wiki
And it has a f/3.3-6.4 zoom lens which might not support very fast shutter speed under not the best lighting condition. For FZ300 which has the same size sensor of TZ80, but has a much faster f/2.8 zoom lens would be better for the job.
Finally can the AF system of the camera keeps up with the movement is another concern. CDAF of compact camera might not track fast moving object as good as PDAF cameras (DSLR). It is a major reason why we have cameras specifically for sport shooting.
As long as we can shoot within the limitation of TZ80, it should be a very nice, compact, full zoom range camera for general usage. For specific usage we might need something else I think.
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Albert