Geoff's diagram needs some explaining:

DarkBlue = early apex, note how it touches the the inside kerb earliest. However, you've only got 20-30 degrees of turning done by that point with a lot more to go. So on the 2nd half of the turn, you have to slow way down in order to make the remaining 60-70 degrees of turning. It also throws you into oncoming traffic.

Green = middle apex with perfectly symmetric entry and exit lines. Halfway through the corner, you've done 45-degrees of rotation and will do another 45-degrees on the 2nd half of the turn. This is the way to go around a corner with the highest average speed.

LightBlue = late apex. This requires a later turn-in and gets 60-70 degrees of turning done by the 1st half of the corner. However, the diagram isn't accurate for biking in that you don't let up on the turning in the 2nd half. It's good for motorcycle & auto-racing because letting up on the turning radius gives you back traction that can be used to accelerate earlier in the corner. This lets you achieve higher top-speeds on the next straight away. However, for cycling, you actually continue the same tightest of turn on the 2nd half of the corner and end up in the middle of the road or the yellow line to stay safe.

For non-powered vehicles like bikes, it's more like this:



The dark-blue line here is more representative of late-apex corners for a cyclist. The late turn-in requires a swift dive into the corner; going from upright to full-lean very quickly. Then holding the turn constant after the apex to come out on the centre to stay safe.

For the OP, I'd practice the swift turn-in in a parking-lot. Learn to relax and turn-in LATER and QUICKLY. Do S-turns and figure-8 turns in a parking lot. Once you get the feel of moving the bike and your body around swiftly and easily, it'll translate into faster cornering at higher-speeds.