My rav4/2002 makes noise when turning (left or right) but the noise will disappear if you tap on the break paddle.

Usually comes from a front or rear wheel bearing.

Replace the right side wheel bearing. But before you replace it. Listen to see if the noise is coming from the front or back. If you can't tell, Put some one in the back seat. Front bearings usually make noise. rear bearings, the noise can come and go.

Probably too late now, but with new pump, jack up front end and put on stands. Fill reservoir to normal full, start engine and slowly turn wheel right and then left to almost lock. Refill as needed and repeat until all bubble disappear from reservoir. Still got noise? Replace pump.

If the problem is as bad as it sounds, you may be close to parking it until fixed.
Normally, if you turn left, your are "loading" the right side, and visa-versa.
Think of it this way.
Picture a car going fast around a turn. The wheel on the inside of the turn might actually lift off of the ground, it is "unloaded".
With that thought in mind, look closely at the right front wheel. Start with the brake pads and /or rotor ( what the brake pads rub on to stop the car.)
Check the rotor inside and out. A good flashlight will do.
Pull the right front wheel and see if the pads are worn out or the rotor is no longer smooth, but "torn up" on the inside or the outside.
If the pads and the rotor is fine, next look at the wheel bearings.
This will be a bigger problem, probably a shop needs to check this out.

I would check the wheel bearings on the left side first. Just for a dummy check make sure that the cowlings to the wheel wells have not fallen down and rubbing on something. They have plastic tabs that sometimes break and leave something hanging.

you are right about the bearing going to bits, it is called a CV joint. it allows the driveshafts to the front wheels to flex as you turn left or right and one is coming apart if one is going bad the other side will too soon, get them both replaced. If you don't, when it breaks completely it will cause MUCH more damage

Check wheel bearings, most likely front 1 or both. get up to speed and take some corners, if it quits down when turning right, the front right wheel bearing is bad. If you turn left and it quiets down, then its the left front thats bad(important to maintain 30-50 mph). They are sold as the whole hub. Also check your front tires, run your hand across the inside and outside edge and see if they are worn uneven, this can make a lot of noise if the tires are bad. if thats the case, have someone check the ball joints and tie rods, if they are bad, it will do that to the tires, thus the unevenly worn tires making a paddling noise effect.

Noises from axles can be really deceiving. The sound is transfered throuh the transaxle and can resonate through the entire vehicle. The turning to the left thing makes it sound like a problem in the right wheel/axle area. Wheel bearings and axles tend to make worse noise when they are on the OUTSIDE of the turn. More of the vehicle weight is shifted to the outside due to centrifugal force. Thus, if the noise is louder when turning left, this puts the right axle to the outside of the turn and the most likely to be the source of the noise. (and vise-versa) The fact that the noise calms down while cruising makes me believe that it is not a wheel bearing. Wheel bearings can make more noise in turns, however, they usually make a "roaring" or loud "humming" noise that gets louder with speed. An axle is more likely to change or get louder according to engine load more than rotational speed. (Way worse while accelerating or climbing a hill) I would be taking a hard look at the right axle.

i woul check the cv joint .then the break.check both side .

Sounds like the break rotors are warped. At this point its best to do a break job, chanses are the calipers are the problem not releasing and creating heat, or break lines are not returning fluid back to resavor witch in turn warp the rotors. Run the whole gamet on the breaks. New rotors, New calipers, New pads, and New rubber break lines to calipers. And you will solve your break problem.