First up 'Sharpness'... oft mooted, seldom particularly well understood, A~N~D why on earth is this your comparison criteria? Was the picture of the thermometer particularly pleasing? Would ANY camera have made it much if any more so?
Oh-Kay... onto some mechanics of the cameras. According to a quick google, that has a 14Mpix not quite a micro sensor, (Its a 1&1/4" diagonal from specs) and an f3.1/5.9 max aperture. Which is actually not so tardy; your Cannon Kit 18-55 should be f3.5/5.6, your Tammy, f3.5/6.3...
Anyway, the F-Number, is a dimensionless unit, the ratio of effective aperture diameter to the lens focal length. If a lens is a 'zoom' that is has a variable focal length, then for the same hole diameter, as the focal length is zoomed, so the F-Number will get larger. Now, the F-Number effects the Depth of Field, or the front to back .. no not 'sharpness'.. but FOCUS.
In essence, its a bit of a swizz, small sensor and mega zoom camera makers use, to flatter their offerings, a bit. The smaller sensor means that they don't have to make a lens with such a huge range of travel to get the same effective 'zoom'. If you have, cos its the old defacto standard, say a 50mm focal length lens; to get a 3x 'zoom' range, the optical; elements inside would have to 'travel' the equivalent of say 25mm to 75mm, in focal length, or two inches. If you shrink the sensor, from 24x36mm, the frame size of conventional 35mm film, to say 16x24mm, the nominal frame size of an APS-C sensor, then enter the 'crop-factor' in this case approx 1.5, and the smaller sensor lopping more off the top bottom and edges of the image circle the lens projects on it, gives an effective field of view, 'as if' the lens had a focal length 1.5x longer. It doesn't, its still whatever it is, but, it has that 'effect'on the image framing. Make the sensor even smaller, and give the camera a 2x crop factor, or as is the case with your Fuji, even more, you can get an incredible effective zoom range, with very little actual 'travel'.
From quick google your fiju, has a 'real' lens range from 4.3mm to 115mm, something like 25/30x 'zoom', but with only 110mm or 4" of travel, thanks to the crop factor, to give a lens with the same 'effective' framing as a 24-600mm lens, used on a full frame or 35mm film camera. Ie, the crop factor lets then get a 'zoom' range, that is 25x, and would beg 500+mm of physical element 'travel', from a lens with barely 100mm or 20% the physical movement.
Double whammy! Using the smaller sensor, the closest and far focus limits are pulled closer to the camera, and the Depth of Field 'zone' is pushed further away, SO, for the same subject, and the same subject framing the camera will tend to be further away from the subject, and the Depth of Focus greater. This effectively means that pictures are more often going to be in the DoF zone, the focus less critical, and the camera has an easier time, and you get effectively 'sharper' pictures.
And to say it's a bit of a sqizz, is a bit ho-hum... as far as the camera maker goes its all win, because what you, the user, normally hopes to achieve, is nice crisp 'in focus' subjects, and a small sensor camera will to a certain degree help you get this, more often than you don't. Especially on a non-enthusiast point and press camera. Where it doesn't really help, is when you actually want to get a shallow DoF and your subject nicely in focus and have your back-ground going nicely out of focus
Now, to get that focus 'fade' from nicely in focus to blurred and out of focus, you need to mess with the maths. One way about is to move away from your subject. Now the subject is further from the camera, and to get the same effective framing, you need a longer lens or less 'zoom'. Now the closest focus distance, and the zone of critical focus is a lot further from the camera, and focus is that much more critical, and electrickey will struggle more to nail what you actually want to focus on, and there will be less tolerance around that focus range where it will fall in the DoF zone of 'acceptable' focus.
As interesting aside to the topic; in days of yore camera makers attempted to win over photographers from Medium Format, 120 roll film cameras to the upcoming 35mm film format. My 120 roll film camera make 6x9 CM images on the film, with a 105mm lens, that has the 'effective' field of view of a 50mm lens on my 35mm film camera that makes just a 24x36 MM image on the film. The closest focus distance, on my 120 'folder' is about 6 feet, with 'critical' focus extending to I think its 300 feet! And that is with a 'fastest' aperture of I think Its f6.3! Most subjects will fall into that 'range' of critical focus, and it will deliver a wonderful focus fade from nearest focus distance to infinity, around that. When camera makers started to 'push' 35mm film, the nearest focus distance went down to around 2 feet, with hyper-focus from around 25 feet. And to get even THAT range of critical focus, where you might see some 'focus fade' they had to push the f-number from f6.3 to f2, and the focus fade becomes rather more obvious, with very distinct change between obviously 'in' focus to distinctly out of focus.
This is what lead, significantly' to the f-number 'arms-race' of the cold war era, as lens makers strove to make ever 'faster' 50mm pime lenses for 35mm film cameras that might help deliver some of this shallow focus effect, photographers had become used to, and was becoming associated with larger format cameras that necessitated longer focal length lenses.
Tendency that is still with us, camera makers vaunting the shallow DoF and 'Bokah' effects obtainable with larger sensor, MFT, APS-C and full-frame cameras and wider aperture lenses.....
Which brings us around; its all swings and roundabouts, and horses for courses. 'Sharpness' is to start with a very subjective topic, and as much down to subject contrast, as well as 'Focus'. Your smaller sensor 'Bridge' camera has inherent advantage here in that the relatively short 'real' focal length lens will be focusing near infinity at relatively short focus distances giving pretty good focus crispness, flattering what you perceive as 'sharpness' . As you zoom 'in' the F-number will also get higher, increasing Dept of Field, again effectively flattering perceived 'sharpness'.. so its all much to be expected of that type of camera really. But push it the other way, and try for those shallow focus and Bokah effects, then the focus is likely to become very clinical with little to no fade between in and out of focus, to the point that in focus 'subject' detail can appear almost cut and past photo-shopped against an OoF back-ground, and you will start to struggle to achieve 'photographic' effects that display any pleasant focus fade. Now your interchangeable lens, larger sensor DSLR will be starting to show its metal... b-u-t, it's something you stilll have to understand and work for, the camera don't do it all for you.
So... have you wasted your money on a DSLR? Well, depends on what you were hoping to achieve and how much effort you were/are prepared to put into get it,
If you were hoping for more perceived 'sharpness', no effort required, then yes, probably, because that's not how it works. If you want to explore what you can do with an interchangeable lens camera, and a bit of know-how.. then no, you probably haven't wasted your money... but you have yet to get the know how to exploit what you got, and will need to put a bit of effort in to getting what it might do for you. Like I said, swings and roundabouts and horses for courses.
