I had a Fuji superzoom (HS20 - about 35x...I think they're probably on the HS30 or 40 by now. Actually I still have it - haven't been able to give it up yet - it's my insurance in case something happens to my current camera) that I used before getting a mirrorless camera. I quite liked it as it wasn't super heavy but I could do wide angle and have lots of zoom (of course, at full out zoom, hand held was pretty shaky) and a lot of the options you would find on a DSLR.
I used to work in a camera store but it's been a few years so I'm not up to date on the models, but Canon and Nikon had some nice, more compact superzooms...I think Nikon may be the lighter weight/more compact of those two. One thing I liked about the Fuji was that you did the zooming yourself as opposed to the camera button doing it, so it was less of a power drain on the battery.
I went with a mirrorless because I sell my photos online and wanted a better quality/option to print larger. (I have the Canon EOS M - looking back, I'd prob go with a Sony now). I got tired of changing between my 18-55 and the 55-250, so I just invested in the new Tamron 18-200, that way I don't have to lug 2 lenses all day and can still do wide to zoom. Not knowing what lenses you have (or camera body)...maybe an upgrade to an 'all-in-one' lens may do the trick.
Another thing that helped me with the weight of a camera was to get a different neck strap so that I could sling it cross wise and it sits more on my hip. I hate the weight on my neck and the camera bouncing on my chest. Joby makes a nice Ultrafit sling strap for men (I have the women's version for my ahem 'curves' and I love it).