"Whenever I have a private, I play perfectly fine. When I'm in a match it just goes downhill. What can I do to stop this from happening?"
What is your focus? Ask yourself what are you thinking about when you're having a private session, and what are you thinking about when playing a match? I bet your focus is different in those two situations. During your private, you’re probably thinking about a mechanical adjustment you’re working on or executing the play. In both instances, your focus is on something you control.
Where is your focus in a game/practice situation? Is it on how you're playing? Is it the game situation? Are you afraid of what others are thinking? Are you afraid to make a mistake? Are you thinking about what the coach is going to say?
You will notice that most of the things I mention for in-game situations are not in your control.
One way to execute in a game, as you do in privates, is to work on your focus. Your focus is going to range from broad to narrow before each point. When your focus is broad, you’re thinking about the game situation and what you need to accomplish on that point. You’ll process some things: the situation, your opponent's tendencies, positioning on the court, and most importantly, what do you want to accomplish for that point.
Once you’ve processed that information, you have to switch to a narrow focus, and that focus is on executing your task/s. This is where you can increase your success rate. After processing all the game information, you need to switch to only thinking about executing your task. To transition your focus from broad to narrow on the court, use a Cue. The Cue can be a word you tell yourself or a physical Cue like a breath you take before each point. That Cue tells your body to focus on executing the task and nothing else.
If you’re having trouble executing in a game because your heart rate is too high, or your breathing becomes short and shallow, or time is speeding up, or in general, your anxiety has increased past your performance point, then we can take another tactic.
The first thing I would recommend is to step back and take a breath in through your nose, hold then exhale through your mouth. If you need to, take a second breath until things slow down for you. Then think about one mechanical or mental key that helps you execute the task. Just make sure you keep the mechanical key simple and controllable. Keeping a solid base or quick feet can be thought about when making adjustments in-game.
You want to put yourself in a position to execute your task. Any energy spent on thoughts outside of executing your task will take away from your play once your point begins. Your ability to shift focus from a broad game situation to a narrow focus on the actual task is what will take your play from the private sessions to the court, allowing you to perform at your best.