The moment I turn on the fan, the fan starts pumping extra air into the space in front of it and the air pressure there should actually be higher than in the surrounding.
You probably mean that the fan is able to push more air where it wasn't and produce an increase in pressure. That is true, but only when the fan is embedded in a wall that separates the two spaces and the space into which the fan is pushing more air is hermetically closed. The pressure inside will rise and stabilize at value higher than pressure outside, due to rotating fan pushing air inside.
Suppose I have a fan blowing air. By Bernouilli's equation, the air pressure in the stream is lower than the air in the surrounding.
Not necessarily. The Bernoulli equation does not relate pressure at two points of different streamlines, only pressures at two points of the same streamline.
In usual airflows, viscous friction between air layers is present. When such friction is present, the Bernoulli equation does not hold exactly even along single streamline. There is some error as pressure drops along the streamline not only due to increase in speed, but also due to viscous friction. But we can still use the Bernoulli equation to make estimates.
So when the fan starts rotating, its blades accelerate the air that is in touch with them. The air increases speed and by viscous friction, it draws other air layers that are not in touch with the blades. When the fan rotates at constant speed, the overall flow in the direction of the fan is as follows (we ignore the screw-like part of the motion. This pattern could be checked by using light pieces of paper, or dust/mist introduced behind the fan):
The air behind the fan is moving from all directions towards the fan and increases it speed:
↓ ↓ rotating
-> | | ->
| | --->
A -> --> B | | C----> D
| | --->
-> | | ->
Here, due to Bernoulli principle, we know the pressure just entering the blade space p_B is lower than pressure far in the back p_A.
When the air hits the blades, it is further accelerated. However, this process is outside the scope of the Bernoulli equation. The air acquires some kinetic energy from the blades, not from the surrounding air, so at point C, the air is faster than at B, but we cannot use the Bernoulli equation to relate this speed increase to pressure change. There is no definite relation between p_B and p_C given only the action of the blades.
Pressure in front of the fan at point C can be lower than pressure at B, if the space in front overall (at point D far from the fan) is maintained at lower average pressure than space in the back by other means (as is often the case when using internal AC unit, since there is another fan that blow some air out). Then, as the air moves from C to D and then to random places, it loses speed and increases its pressure, but the final value at D can still be below pressure at B.
Or, if the room is hermetically closed as mentioned in the first paragraph, the pressure at D is already higher than pressure at A and then the fan pushes the air inside to stay there even though it has higher pressure than the air outside.