Well... If it really was blowing like that you wouldn't have been able to transition from a close to a Broad reach to gybe without easing the Jib and the main first. You would have a tough time gybing from a close reach

You want to accelerate and have as much boat speed as possible. This will also make the gybe more manageable as all that pressure won't hit the sail as hard when you come around. What I do is often sheet in and 'heat' the boat up a bit to build a little extra speed, then sheet out to help the boat turn. Once the point is nearly pointed dead down wind with the sails out sheet and travel in as you continue to carve out the turn. This centers the boom and doesn't give it the punch it has if let the wind take it across. (you can also grab the boom/mainsheet b/t the blocks and throw it over. Then don't forget when the wind fill that main on the new tack it is going to want to round the boat up.. So unless you are ready for it be ready to pull it back down.
When tacking... Don't let out the main or jib sheet until the boat is head to wind. In fact sheet that sucker in as tight as you can when you start your turn. The leech of the main will help you turn.. Once head to wind THEN ease the main and let the jib pull the boat around. If it won't IMO the first thing to do is drop the main a bit further and pull the jib clew into the wind backwinding it. If you have blown the tack be mindful you might already be moving backwards. So pop them rudders over... and if it still won't go push the main all the way out backwinding it..
Once you figure out the "Y" turn you will find you don't even need to make it totally head to wind to get through.