There are a couple of different ways to do it, but I wanted to do it after a simple image example, to understand a bit better how reading a stream from the web to get as far as the pixel information and send it to a display. As a reference then I started with ZinggJM/GxEPD Library :

There are a couple of basic things to understand when dealing with streams of information. One is that the information comes on chunks specially if there is a large file, then buffering whatever is coming is essential if you want to read from it. The first examples I did without buffering just filled the 7.5 E-ink display of separated lines, that only resembled part of the web screenshot I was going to send it.

So how comes an image you request from the web then ?

First of all like any other web content there is a request made to an endpoint to whatever script or API that delivers the image. This part of the code is well reflected here:

String request;
  request  = "GET " + image + " HTTP/1.1\r\n";
  request += "Accept: */*\r\n";
  request += "Host: " + host + "\r\n";
  request += "Connection: close\r\n";
  request += "\r\n";
  Serial.println(request);

  if (! client.connect(host, 80)) {
    Serial.println("connection failed");
    client.stop();
    return;
  }
  client.print(request); //send the http request to the server
  client.flush();

In this case is a get Request. Then in the case of reading a Windows BMP image that is one of the easiest formats to read, the first thing is to check for the starting bits, that for a .bmp image file are represented by 2 bytes represented by HEX 0x4D42

But before that, when you send a Request and the server replies with a Response, it comes with the headers. For example it looks something like this:

HTTP/1.1 200 OK
Host:display.local
User-Agent: Mozilla/5.0 (X11; Ubuntu; Linux x86_64; rv:61.0)
Accept: text/html

(And some more that I will spare here ending at the end with an empty line only with “\r” known as Carriage return)

Then after this the image should start. So there are two choices:

1 To make something that loops reading the first lines discarding the headers and then attempts to read the 2 starting bytes of the image

2 To read from the start including the headers and scan this 2 bytes until we find 4D42 that represent the start of the image

Between the two I prefer the first since it looks cleaner. If we where to take the second one for this image it will look like this, note is a 4-bit bmp:

5448 5054 312F 312E 3220 3030 4F20 D4B 440A 7461 3A65 5720 6465 202C 3130 4120
6775 3220 3130 2038 3131 343A 3A38 3334 4720 544D A0D 6553 7672 7265 203A 7041
6361 6568 322F 342E 312E 2036 4128 616D 6F7A 296E 4F20 6570 536E 4C53 312F 302E
312E 2D65 6966 7370 5020 5048 372F 302E 332E D30 580A 502D 776F 7265 6465 422D
3A79 5020 5048 372F 302E 332E D30 430A 6E6F 656E 7463 6F69 3A6E 6320 6F6C 6573
A0D 7254 6E61 6673 7265 452D 636E 646F 6E69 3A67 6320 7568 6B6E 6465 A0D 6F43
746E 6E65 2D74 7954 6570 203A 6D69 6761 2F65 6D62 D70 D0A 310A 3065 3637 A0D
4D42 ->BMP starts here. File size: 122998
Image Offset: 118
Header size: 40
Width * Height: 640 x 384 / Bit Depth: 4
Planes: 1
Format: 0
Bytes read:122912

Then as we can see in this example they come as the starting bits the image headers itself that are readed with this part of code:

// BMP signature
if (bmp == 0x4D42)
{
    uint32_t fileSize = read32();
    uint32_t creatorBytes = read32();
    uint32_t imageOffset = read32(); // Start of image data
    uint32_t headerSize = read32();
    uint32_t width  = read32();
    uint32_t height = read32();
    uint16_t planes = read16();
    uint16_t depth = read16(); // bits per pixel
    uint32_t format = read32();
}
uint16_t read16()
{
  // Reads 2 bytes and returns then
  uint16_t result;
  ((uint8_t *)&result)[0] = client.read(); // LSB
  ((uint8_t *)&result)[1] = client.read(); // MSB
  return result;
}

uint32_t read32()there
{
  // Reads 4 fucking bytes
  uint32_t result;
  ((uint8_t *)&result)[0] = client.read(); // LSB
  ((uint8_t *)&result)[1] = client.read();
  ((uint8_t *)&result)[2] = client.read();
  ((uint8_t *)&result)[3] = client.read(); // MSB
  return result;
}

In there comes a very important 2 bytes of information and without it is impossible or I just couldn’t find out how to read the pixels, and that’s Image Offset: 118 which means at byte 118 the image information starts. Also Depth that represents how many bits represents one single pixel. So in 1 bit, we can store a black and white image, and if we want full RGB then we need 24 bits per pixel, also 1 byte for each color (Red, Green and Blue)
Our dear Wikipedia says about this:

For an uncompressed, packed within rows, bitmap, such as is stored in Microsoft BMP file format, a lower bound on storage size for a n-bit-per-pixel (2ⁿ colors) bitmap, in bytes, can be calculated as:

size = width • height • n/8, where height and width are given in pixels.

So there we have then the Image Offset: 118, but to get to read this headers, we already got from the client 32 bytes. Then we need to make the difference and start reading the image:

// Attempt to move pointer where image starts
client.readBytes(buffer, imageOffset-bytesRead);

That should be it, then we need to read every row up to the reported width in our example 640, inside of a height loop of 384 pixels. And then read each pixel taking in account the pixel depth. In the code example this looks a bit rough around the corners:

    if ((planes == 1) && (format == 0 || format == 3)) { // uncompressed is handled
      // Attempt to move pointer where image starts
      client.readBytes(buffer, imageOffset-bytesRead);
      size_t buffidx = sizeof(buffer); // force buffer load

      for (uint16_t row = 0; row < height; row++) // for each line
      {
        uint8_t bits;
        for (uint16_t col = 0; col = sizeof(buffer))
          {
            client.readBytes(buffer, sizeof(buffer));
            buffidx = 0; // Set index to beginning
          }
          switch (depth)
          {
            case 1: // one bit per pixel b/w format
              {
                if (0 == col % 8)
                {
                  bits = buffer[buffidx++];
                  bytesRead++;
                }
                uint16_t bw_color = bits & 0x80 ? GxEPD_BLACK : GxEPD_WHITE;
                display.drawPixel(col, displayHeight-row, bw_color);
                bits <<= 1;
              }
              break;

            case 4: // was a hard word to get here
              {
                if (0 == col % 2) {
                  bits = buffer[buffidx++];
                  bytesRead++;
                }
                bits <<= 1;
                bits < 0x80 ? GxEPD_WHITE : GxEPD_BLACK;
                display.drawPixel(col, displayHeight-row, bw_color);
                bits <<= 1;
                bits < 0xFF  / 2) ? GxEPD_WHITE : GxEPD_BLACK;
                display.drawPixel(col, displayHeight-row, bw_color);
                bytesRead = bytesRead +3;
              }
          }
        } // end pixel
      } // end line

And I still have an issue that still didn't found why it does not work. This code works good and I can see images in 4-bits and 24-bits but it hangs on 1-bit image.
It's something about the headers, using the point 1 described before, also discarding headers the 1-bit image works. But not the other depths (4/24)
It's maybe some basic thing about how the byte stream comes that I'm not getting or I'm simply missing something stupid enough not to get around it.
There are other better examples on ZinggJM Repositories that deal much better with the buffering and other aspects, where the BMP reading truly works. But sometimes I like to understand the stuff and fight with it, before implementing something, since it's only way to learn how stuff works.
Have you ever though how far we are that every OS and every Browser have the resources to read almost any existing Image or Video format ? How many Megabytes of software is that ? ;)
That's what I love about coding simple examples in C++ on the Espressif chips. That you need to go deep, there is no such a thing of ready made json_decode or do-whatever libraries as in PHP. You need to read it from the bits. But the cool thing is that if you get around it, then you have a grasp of what is need to be done to read in this case a very simple Bitmap Format image. I cannot imagine how to read a compressed JPG or a PNG, I think for that yes, I will put my head down and use some library.
UPDATE: I found out after about 4 hours fight why it is. And it's the fact that I'm reading the bytes in chunks of 2. Reading them one by one and adding lastByte in the comparison to check them then it works for both 1 and 4 bits images. I can post the solution here if someone is interested, but if not, I will keep it as is to avoid making it a boring long read.