I need this for my runs

There might be a specific type of technique to it, but the way I do it is I do a dodge (move with a or d, rarely s and then press shift), then you turn to face the direction you're dashing and let go of a/d/s and run forward with W. It might take some practice to get down

and let go of a/d/s

What's the benefit of this?

There might not be a benefit, it’s just become muscle memory for me at this point. Figured that it would make you faster going straight, but it might not matter

I've been practicing this for a while but I can't seem to get the big burst of speed like the other runs do that make jumping from the apartment roof to the bridge possible. What's the secret here?

the bridge jump is a pain in the ass honestly, you gotta be right on the edge (thats how I did it for my run). Your dashes are probably fine, its just that jump particularly sucks

It really just takes practice. What I would recommend doing at first is binding your sprint/dodge key to a mouse button (I have mine to one of the buttons on the side)

Start by just moving with both forward and a strafe key then release forward and sprint before re-pressing sprint then forward in quick succession. essentially dont worry about releasing strafe at first.

Doing it this way is still enough to make the jump in the first level and is a lot simpler to both comprehend and physically do before working on releasing strafe for the faster sprint.