You could try the optimised PDQ library here. That will probably run twice as fast but if you need better performance then you will need a faster processor, for example a Due plus a parallel port display.
I have already make a project using the TFT.h library and some functions are not supported with the Adafruit_ST7735.h (for example : screen.text(text, xPos, yPos); screen.loadImage(name); etc.)
If I could make no change of my previous code it will be great.
Is there another way for increase speed? I think Esplora was using the same screen with an ATmega32u4 (that's run at 16MHz too).
There is no need to change your board or the display.
The SPI interface should give you a fast snappy display. And is considerably easier than all the wires and level-translation of a parallel display.
Yes, the basic Adafruit_ST7735 library could be faster.
I was not aware of the "PDQ_ST7735" library. I will see how it compares with my own one.
What bitmap is taking too long?
A small icon in Flash memory is very quick.
Processing a 24-bit BMP file on a SD card is reasonable.
If you pre-process the 24-bit BMP into a 16-bit RAW file on the SD card, it is less bytes, less processing and faster to draw.
Most BMP examples use 24-bit. There are several other BMP formats that have smaller disk files and less processing than 24-bit. But your sketch needs to accept these other formats.
It seems that I have to change my board for a fastest one.
Before you part with your money and end up being disappointed it would be worth stating what performance you are looking for (full screen bitmap updates per second?), indicating your operating scenario and why this high performance is desirable in your project. Often very significant performance increases can be made simply by making small changes to the sketch (for example only updating areas of the screen that change, example here). You may find no standard (non-optimised) setup will meet your needs even with a Due, so for any performance improvement path some code mods/hacking will be needed (which you appear reluctant to do).
Explain what you want to do. And at what speed you want to do it.
You can certainly render a photo from a SD card. Or draw graphics from scratch.
The PDQ_ST7735 library is very impressive. (I am not so impressed with the examples)