So we have Bernoulli's equation which tells you about the change in different types of energy when fluid flows from one point to another. I can understand the conversion between kinetic and potential energy but the conversion between kinetic and pressure energy is not clear. I thought pressure it self was an effect of the kinetic energy of moving particles. How can then we increase the kinetic energy while reducing the pressure
There are two components to the kinetic energy of the fluid particles, the first one is that relative to the center of mass of a fluid element. This velocity components are responsible for the random motion of the molecules inside a volume element but produce no net motion of the fluid itself. This motion is also responsible for pressure and temperature. This motion also averages to zero.
The second component of motion is the motion component due to the movement of the fluid element itself (just imagine that in the first example the fluid was at rest, and now it starts to move as a whole), which is responsible for the macroscopic motion of the fluid. If you average the velocity of all the particles and is different than zero then the fluid element itself has to move. This velocity component results in a macroscopic motion (you can see the fluid moving), it is not microscopic. The kinetic energy referred to in the Bernoulli's equation is only the macroscopic component, that is, the motion of the fluid as a whole, regardless of the internal motion details.
The relationship of this macroscopic motion to pressure is due to the following:
"if a fluid is flowing horizontally and along a section of a streamline, where the speed increases it can only be because the fluid on that section has moved from a region of higher pressure to a region of lower pressure; and if its speed decreases, it can only be because it has moved from a region of lower pressure to a region of higher pressure. Consequently, within a fluid flowing horizontally, the highest speed occurs where the pressure is lowest, and the lowest speed occurs where the pressure is highest" (Resnick, R. and Halliday, D. (1960), section 18-4, Physics, John Wiley & Sons, Inc.)
The pressure in Bernoulli equation is different from the "pressure" in gas. The pressure in Bernoulli equation is due to force applied to the surface of the fluid volume. The force can be, for example, due to gravity. The pressure is not a function of temperature as in gas.