FSC certification is basically an auditing scheme that aims to trace timber from tree to end-product. Most timber imported for sale via larger timber merchants will be imported as part of this or similar schemes, but smaller makers like myself are not able to claim FSC certification unless they spend huge sums to become part of the scheme themselves.
These schemes primarily seem to be intended as a means for the mass-producers to indicate that they meet basic environmental and welfare requirements, especially if they want to bid for large contracts.
I have been told by one or two smaller timber merchants that even if they were to convert a tree from a wood half a mile away and sold it to a maker in the same town that they would not be able to claim any certification level because it too expensive for them to join the scheme.
So, while it is essential that these schemes exist, they should not be considered as the total answer.
In Europe, the EU Timber Regulation (EUTR) prohibits the placing of illegally harvested timber on the European market so I would probably be unable to buy imported timber from an illegal source if I wanted to. As with so many aspects of life, however, being “legal” may not quite the same as being “moral”. The FSC scheme does also provide some protection for biodiversity and indigenous populations.