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I was hoping someone could help, are there any relatively good graphics cards which will fit this narrow PC case?

Not exactly what you're asking but if you go to Amazon (or similar), search for graphics cards and tick "Low Profile" it'll show you what will fit in.

Do you have a graphics power connector spare?

I don't think that it's a low-profile that's needed as the actual bracket which holds the graphics card in, seems to be half the high of a normal one?

You'll need to be sure of the slot type too. That looks to be a fairly (tech-wise) model you have there, and slots may have moved on.
Check the spec of the machine, as it should list the available lots.

Seems I need one of these: https://www.ebay.co.uk/itm/GTX-1050-Low-Profile-Bracket-for-Video-Graphics-Card-DVI-HDMI-DisplayPort/113201117819?epid=22022565205&hash=item1a5b4ffa7b:g:ZcMAAOSwtAtbn1pP:rk:1:pf:0

Do you think it'd be hard to change the bracket?

EDIT: It's got an i7-2600K and PCI-E so I don't think it'd be an issue

That bracket is for a specific low-profile graphics card (the GTX 1050 comes in standard and low profile types, and presumably the same is true of other makes and models). You'll need a low-profile card to fit in that case, and then make sure that either it comes with a low-profile bracket in the box (as the Serial card I bought recently did) or you can get the appropriate low-profile bracket separately (that one you linked to won't fit the example in this post; you'd need this one).

How old is that PC? There don't seem to be any USB-3 ports; that and the presence of PS/2 ports for mouse and keyboard would suggest it's an older model (and the processor was introduced in 2010).

Do you have a use for the card set yet, or you just want to upgrade to what you can?

Is the current setup a bracket that attaches to a motherboard port, or a PCI-E card? If card, what make and model is it?
If you go for a 1050 or similar, they're pretty power-hungry, and might need a certain type of power cable to work properly. Check what size your PSU is before splashing out on a GPU card.

Have you tried PC Part picker?

https://pcpartpicker.com/products/video-card/

You can make it show you cards of different thicknesses and lengths so if you measure your old card you should be able to put in similar dimensions and it'll find ones that will fit in your case