I wondered something similar about Timberlake's hair when I saw a recent pic of him. Definitely not the curly mop I remember from my teens.

It looks like it's been flat-ironed, chemically straightened, greased and then combed into submission. OP I think you could try a BKT system, those are supposed to be temporary. I would just make sure to read up on their ingredients and use one without formaldehyde. Go with a relaxer if you want a permanent option...and remember to consider the upkeep of these options.