Can I upgrade this BNC security camera system without rewiring?

I have an older Speco camera system and DVR.  I'm guessing circa 2008 era.  The problem is, it's laughably difficult to access, fix settings, or even watch/review video footage.
Priority 1 - Change the DVR to a new DVR or NVR that allows real web access, cloud, app control.
Priority 2 - Upgrade the cameras to newer, better, resolution.
I can not rewire.  It's a big, modern house.  Stucco.  Etc.  It wouldn't be worth the cost, hassle or benefit. 
Can I just *yank* the old DVR and input a newer one with better tech?  Is everything POE now?  Do I try to convert the BNC to POE?


There are a few options.  You can use converters so you can run IP cameras over coax, but they are expensive.  HERE is an example.
I'm a little confused when looking at your camera picture as it shows multi-conductor cabling, yet you stated you had coax running to your cameras.  Do you have AC power at your camera locations?
How is your WiFi coverage around the outside of your home?  You may consider using the coax cable to run the 12 volt DC power most cameras will use (small currents) over the coax and use WiFi for their Internet connectivity.  Of course you would have to select WiFi capable cameras.
I use a Synology two drive device (mirror RAID) with their camera surveillance software for my DVR.  I went this route because I also wanted a reliable backup/central storage for my home computers.
There are plenty of options depending on your home's infrastructure capabilities.  Just some thoughts...
If you do have coax cabling to the cameras there's a POC (power over coax) solution that Hikvision and some other manufacturers have marketed.  They make DVRs and cameras that implement the solution and also can use IP feeds from POE cameras.  I have one.  Works great.
Thanks for all the input thus far.  Tbh, it's getting more confusing by the moment.  
So I called Speco.  They won't talk to me on the phone.  The answer my emails but they don't answer my questions.  So frustrating.
I replaced the DVR with a newer Speco DVR.  It works, well enough.  But the camera quality is still garbage circa 2008.  
I'm trying to upgrade the cameras using the existing wiring.
On the back of the DVR, the connections were all BNC.  But on the cameras, it's some sort of twisted, colored, thin cable.  I ordered the camera that Speco recommended and of course, it's not the same output.  Their camera output is BNC not the twisted, thin wires.  
I'm really stuck here.  Any suggestions? 
Thanks for all the input.  I figured it out.  I tried a new camera with better resolution.  It wasn't worth the time to bother.
Better resolution still equally pretty crappy picture.
So even if I spent the time and money to install, I would just be giving the police a better, poor quality picture that they will also never use to solve anything.  lol. 
Your old BNC cables most likely also provide power to the cameras via the Speco DVR.  While removing the old camera did you see some sort of device / box which the BNC cables where plugged in to?  (thinking all of those questions have been answered above).
An upgrade that would work would be to use the BNC cable only to provide power to the cameras and use Wireless / POE cameras.
Typically if you can run CatXX cable then you utilize POE cameras.  That said with a good Wireless Access Point or maybe a dedicated WAP you can go with 4k wireless cameras.  There are a bunch on the market...all inexpensive these days.  
Look on Amazon and search for wireless 4k outdoor video cameras.  I have my 4k video doorbell connected wirelessly to the wireless network and it does work OK for me.
Here utilize Ruckus Wireless Access Points.  You can get a inexpensive DVR which only works with network cameras that would work for you.
This solution would utilize your cables for your wired SD cameras.  You could try your existing wireless access point then maybe later upgrade it.  You could also try your new Speco NVR if it works with networked cameras or purchase a network only NVR.
I just went through an upgrade of my DVR/Cameras using the same coax wiring. Look in to "HD over coax", also called HD-analog or HD-TVI.  The Digital Watchdog VMax A1 is super. I really love it over my old DVR, and it was not too expensive. You can find Digital Watchdog products  at B&H Photo/video. I have found them to be very reliable.
Hmm, I think it’s not the best idea. It’s both non-promising and still hard to do. If you don’t like the rewiring variant, then I think you should consider buying a wireless security system. It’s the best way to solve your problem for me
NagaHuiags said:
Hmm, I think it’s not the best idea. It’s both non-promising and still hard to do. If you don’t like the rewiring variant, then I think you should consider buying a wireless security system. It’s the best way to solve your problem for me
I got such a system, which is much better than wired ones. I’ve got Vivint doorbell camera pro and other smart devices from this company. However, I can’t tell you more details about the installation because I ordered it from their service worker. I could do this, but it would be a hustle and a precious time spent on work I’m not getting paid of. I try to work as much as possible and spend free time with my kids and wife rather than installing some systems.