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I just cannot get faster at any puzzle.

This is definitely false. Chances are it's not your turnspeed, but some combination of efficiency and lookahead. People can get well under 20 with slow turning.

But to answer your question: drill algs mostly. Turning as fast as you can in F2L is almost never a good idea
 
I used to use Z4, and it really is nothing special. (very inefficient)

I strongly recommend a ZZ-adaptation of the Yau method. Basically,
1) first 2 centers
2) form and place df and db eoline "cross pieces" as well as another random edge pair on the left
3) complete rest of centers and another random edge pair on the left
4) pair edges (try pairing the last few edges intelligently to orient some previously made edge pairs [RU'R' vs R'FRF'])
5) orient the last few edges (that you didn't orient before) and solve orientation parity
6) zz it out + pll parity
 
I used to use Z4, and it really is nothing special. (very inefficient)

I strongly recommend a ZZ-adaptation of the Yau method. Basically,
1) first 2 centers
2) form and place df and db eoline "cross pieces" as well as another random edge pair on the left
3) complete rest of centers and another random edge pair on the left
4) pair edges (try pairing the last few edges intelligently to orient some previously made edge pairs [RU'R' vs R'FRF'])
5) orient the last few edges (that you didn't orient before) and solve orientation parity
6) zz it out + pll parity

Thanks I like that a lot. But which centers should you do first?
 
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