How does the sway control work in an Equal-i-zer hitch?

THe location where it sits now (where it's welded) looks like the bar will slide out again on sharp left hand turns.

Something doesn't look right with your installation. The L-brackets should fit flush against the link plates. The nuts on the link plates should be on the inside.
The two 1/2" x 3-1/2" bolts should fit into the pocket on the outside of the link plate.
You might have the outside link plates backwards, I can't tell in your picture. That would cause the heads of the bolts to prevent the L-brackets from fitting flush.

I have been lubricating the "L" brackets on my Equal-i-zer hitch for years now without and noticeable loss of sway control. It The lubricant suppresses but does not quiet the hitch noise when making sharp turns. A dry lube leaves less of a mess and doesn't attract dirt.

The system really doesn't have any real form of sway control! Look at how it's designed. It's basically a friction bar. The EQ statement hat he pivot points at the hitch head are the main sway control are also questionable. It would take quite a bit of angle on the hitch for them to do anything, and that control would be after the fact. The DC has the detents in the bars that increase pressure as the move, and once it stops it will want to return back to center. The DC is the better system. The EQ doesn't offer any sway control compared to the DC.

Do not lubricate the L bracket. That will reduce the friction which will then reduce the sway control. Friction, not lubrication is how it works.As WYNBob pointed out, the L bracket appears to not be sitting flush to the bar, could be bent. By the wa6, those pads help quiet things down a bunch.

See the shiny streak on the top of the L barcket, it is not clear across, indicating it is not sitting flat as Barneys said it should be.

The rusty area is where it usually sits.

The upward pressure on the underside of the hitch head from the WD bar socket tension is the main sway control. The downward pressure on the L brackets from the WD bars is the secondary point of sway control.

You can lubricate the bars sockets where they contact the hitch head if you wish but it will reduce the amount of sway control a bit but will also reduce the noice from the hitch when turning.
Same goes for the L brackets although Equal-i-Zer sells a pad to put on the L bracket to reduce noise.

Most people just put up with the noise as it lets them know the hitch is working.

As far as the brackets moving goes - they should not move if you have the bracket set so the bars rest squarely on them and are not at an angle. The bars should slide smoothly and evenly on the surface of the L bracket. If the bars are angled upwards then they will contact the front edge of the L bracket and possibly cause the bracket to get pushed backward during a turn. This condition will also reduce the amount of sway control a bit.
Barney