The problem with the above setup is, using the image as mix factor directly or with the help of a Color Ramp converts the colored image into a greyscale image which gives a combined brightness value of the overall color, although separating the image with a Separate Color or Separate XYZ node you can see, the RGB channels might be very different in brightness like shown in the following comparison:
As to be expected with yellow/orange flames, the red channel has the most bright pixels overall, the green is still quite bright while the blue channel is mostly dark.
So the secret to keeping the brightness and detail of the original image lies in separating the channels and using them for Emission shaders with different colors and adding them to a transparent material.
To achieve this, you first of all need three Add Shader nodes. Plug a Transparent BSDF into the first Add Shader. The Image Texture will be plugged into a Separate Color node (or Separate XYZ, it does not matter since the RGB channels are a 3-dimensional vector as well).
Now you need three Emission nodes: the first one set to pure red (RGB: 1/0/0), the second to green (RGB: 0/1/0) and the third to blue (RGB: 0/0/1). In the Strength inputs you plug the Separate Color outputs of the according color channels: Red into the red Emission shader, Green into the green one and Blue into Strength of the blue Emission node of course.
Now plug the red Emission into the Add Shader with the Transparent BSDF already plugged in, the output of the addition into the next Add Shader together with the green Emission and that output in the third Add Shader together with the blue Emission. The order does not matter though, since addition is commutative.
Now the result without improving anything like e.g. boosting the brightness already looks a lot better than the basic transparency setup in the question, the colors are more vivid and there is more detail in the flames:
For tweaking the look there are now different options, basically depending on what you want to do and which method you prefer. Just for a general increase of the emission strength overall like in the question you can use a Vector Math node set to Scale and plug it between the image and the Separate Color node:
But if you are maybe not happy with the color of the flames, let's say they are to yellow and you want them a little more red, and maybe going towards white in the bright areas - in this case, instead of scaling the channels uniformly, you can set the Vector Math node to Multiply and use different factors per channel. So for a little more red than yellow, multiply red/X with a slightly higher value than green/Y. For making the bright areas turning more white, multiply blue/Z with a higher value than red and green:
Another way to change the strength of the emission with also having control other the individual colors would be to not scale the image before separating the colors, but multiplying the separate channels after splitting them up with normal Math nodes set to Multiply after the Red, Green and Blue outputs.
If you want to increase the contrast you could either do this with a Bright/Contrast node plugged between image and Separate Color node or for each channel separately by using Math nodes set to Power. Plug the color channel into the Base input and set an Exponent. Exponents > 1 will increase contrast, values between 0 and 1 will decrease contrast.
In the above example I scaled the strength before separating colors, however you have to keep one thing in mind: Let's say you want to increase the strength by 5 and do this with a Scale node. Then you want to increase the contrast by an exponent of 2 maybe, but: it makes a big difference if you first scale by a factor of 5 and then use the Power node on separate channels or if you do not scale before separation, then use the Power node and afterwards a Multiply node with a factor of 5. In the first case your maximum strength suddenly jumps to 25, in the second case you keep the maximum strength at 5 as it was before changing the contrast.
The mathematical reason is simple, in the first case the strength per channel $y$ is calculated as
and since $x\in[0,1]$ this results in $y\in[0,25]$, while the second case is calculated as
$$y=5x^2 =5\cdot x^2$$
and so for the strength you get the range $y\in[0,5]$.
Here you see the first case:
And this is the result of the second case: