[ice445]

Squeaking noise during turning is probably dry tie rod ends or ball joints, nothing related to the PS pump. The easiest way to bleed these pumps, besides lifting the front end and going lock to lock like 50 times (never stay at lock more than a second), is to buy a mityvac and an adapter cap for ford power steering pumps. Then just pull a few inches of vacuum, start the car, and do the same as before, but you'll notice that it takes less than a minute this time before the system quiets down.

 

[ice445]

This is what I thought. But the noise appeared right after I changed my P/S pump today, so that would be an (unlucky) coincidence.
For the dry tie rods or ball joints, can I apply so lubricant to it to reduce the noise?
I do however also have the P/S that is very noisy, so there is also definitely air in the system.

You need a grease gun with a needle tip attachment since they are sealed boots with no grease fittings. But yes, you do definitely have air in the power steering system based on the symptoms. https://www.amazon.com/Mityvac-MITM...51-bdde-775e2ae28371&pd_rd_i=B015XUH31U&psc=1

That kit plus a hand pump mityvac will get you sorted if you don't want to spend tons of time messing with this.