Can I install an upright handlebar (curved towards me) on a touring bike?

If the currently installed stem is relatively long you could also solve this with a shorter stem.

You should also check your saddle position. Most of your weight should be supported by your feet, some of it by your butt and only a relatively small amount with your arms/hands. If your saddle position is bad it can make it hard to keep weight of the hands.

Although that bike has "touring" written on it, I wouldn't describe it as a touring bike. To me (and, I think, most cylists), a touring bike is a drop-handlebar bike similar to a racing bike, but with more relaxed geometry, mounts for racks and fenders, and longer seat-stays so that your heels don't hit your panniers. Typical tourers have reasonably narrow tyres (25mm or maybe 28), whereas yours seem much wider (35mm+). I would describe your bike as a city bike or hybrid.

The seat tube seems to be very leant-back, to me, which gives quite a long reach to the handlebars. Your stem is also pretty vertical, which means you don't have many options for bringing the bars closer with a shorter stem. You might get some mileage out of moving the saddle forwards on its rails, but that will also change your position with respect to the pedals. You could try a taller stem. Fitting different handlebars won't make any real difference, because you already find it hard to reach the flat part of the bars you have, and those will always be the closest part. Check with a bike shop to see if there's anything they can do for you.

Fundamentally, it looks like your bike doesn't fit you, and I don't see a lot you can do to make it fit better. And if you're not comfortable even on a ride as short as 5km, I'd say your bike is a long way from fitting you. Ultimately, I think you're looking at replacement.

I would get another hybrid, but one that actually fits you. Make sure you take it for a test-ride before you buy. That should be able to give you a reasonably upright riding position (make sure the shop knows that's what you want!) in a bike that doesn't weigh a ton and has a decent number of gears. And it won't be any more theft-prone than your current bike. I'm not sure what kind of dollars you're talking about (presumably not US dollars, since you use kilometers) but I'd expect to get a competent hybrid for about £400. Your current bike looks like it's in good condition, so you'll be able to recoup some of that by selling it.

You could also look at adapting yourself to your bike, as well as adapting your bike to you. In the context of road cycling, people often recommend yoga and core strength exercises to help get you more comfortable in a lower riding position – I assume the same would apply in your case. Also, if you're carrying stuff in a rucksack, you might be more comfortable with that load in panniers on the racks your bike already has. And cycling with a full belly makes getting low uncomfortable.

If you are living in one of the larger European cities you have the option of bike rental schemes that rent out decent city bikes. That is, not the vandalism proof kind of 'Boris Bike'. Some work on a daily rate with price caps for weeks or months. Others on fixed monthly rates.

For example, one of my family rents a good 7-speed Dutch bike for €18 a month from a Dutch start up company. In three years that accumulates to about the price of buying new an equivalent bike.

Such rentals may give you an idea what you really need. Or they may even obviate the need to own a bike.

You could consider rotating your handlebars backwards so the bar-ends are facing you. You could keep the shifters/brake levers in the same position as they are now but this would be a disadvantage since you will have to change hand positions to be able to reach them (especially problematic when you have to brake suddenly) assuming your hands are on the top of the bars and not near the shifters/brakes. As an alternative to the current brake/shifter configuration you could consider using bar end brake levers and install them on the top of the handlebars so you can easily reach them from a higher up (on the bars) hand position. Bar end brake levers look like this:

I have photoshopped the image in the original post to resemble what it looks like to 'turn the bar ends towards you'. (I am not 100% what the original poster of this answer meant with his answer so if the picture doesn't match his idea his/her input would be greatly appreciated, please let me know).

You could also use older style drop bar brakes and install them on the top end of your bars if there is enough room for them. It won't look as good but is a cheaper solution. The brake type I am referring to:

The shifters could remain in their original location since they are less important than the brakes (if you can't reach your brakes you can crash, if you can't reach your shifters it might be inconvenient).

If you want to move your shifters upwards as well you could consider using friction shifters such as shown in this image (Original post: Reversing Drop Bars)

Using dutch style handlebars is definitely an option but depending on your needs you might still need to increase the stem height (adjust it if there is extra length left or replace it if it's already at or near its extension limit). The shifters/brakes could be installed next to the grips on these handlebars (as they usually are): (the bike in the image has no shifters but it is pretty much the same, place shifters next to brakes in same location as shown in the image)

To potentially resolve the issue pointed out in the comments by @Criggie♦ of the pipes sticking towards you which could be dangerous if you happen to fall onto the handlebars in a crash for example.. you could perhaps install bar end caps to at least lessen the risk of injury. These bar end caps come in two types in my experience, the type which you press in and the type with an expanding part which clamps the bar end cap in place when you tighten the bolt. I prefer the latter. They look like this: