Hello @Presiyan Kisyov
Thank you for your question. I'm glad to be able to address this topic cos the lord knows it's the most asked.
When we use the phrase "singing high notes", it could mean multiple things because any sound has additional qualities to it besides the pitch ie., volume & tonality.
The 'easy' natural way of singing high notes is to access and sing in your pure head voice. Your vocal coach might have addressed this but you can achieve this by imitating a yawny voice and saying "hoo" like an owl. The volume maybe not as loud and the tone of the sound would be dull and hooty. Nonetheless, it is still considered 'singing high notes' and many singers use this co-ordination to sing some incredible music that's out there.
On the other hand, what you maybe referring to is singing high notes with a considerably loud volume with a bright buzzy tone. We would also call this 'belting'. Many struggle with belting because they try to force this sound out by pushing and tensing up the throat just like you mentioned. This is dangerous for the voice because every time you repeat this mistake, you simply teach the body to do it over and over inadvertently every single time you sing high.
First advice I would give you to achieve your desired goal is to stop doing it the way that you are doing it ie., pushing. Your muscle memory needs to 'unlearn' this harmful technique so that you don't damage your voice permanently.
Secondly, if you want to sing high powerfully, you need to work on singing it lightly first. Because it'll help you vibrate the vocal cords to the pitch you need it in without all that tension. Look for head voice exercises. Remember to keep your voice relaxed, yawny and hooty when you do the exercises (that goes for your lip trills too). Another useful exercise to develop your head voice is the mums. Sing 'mum mum mum' on a scale with a dopey 'duuuh' voice. Make sure your voice remains in this dopey mode through out the exercise. Build this co-ordination by spending some time with it.
Now, once you can sing in your head voice comfortably and with good control, we can start addressing the tonality.
You want your high notes to sound like your chest voice - nice, buzzy and powerful. You can do this by accessing your pharyngeal cavities to make this buzzy sound (NOT your chest voice being pushed up. ouch). Make a nasty whiny nasal voice sound and say 'eh'. Like Donald Duck or Bugs Bunny. Notice how you don't need to push to make such a sound. Imitate those sounds. Get used to making that sound nice and buzzy without using any force. Think like a comedian instead of a singer. Speak in that voice and get used to it. Make fun of your roommates during this quarantine using that voice. :)
Once you think you are comfortable with it, you will be combining the two skills your body would have learned over time to start developing your belt. A good exercise to build it are the Nay Nays on a scale while using that nasal buzzy sound. Make sure you are adding a hefty amount of whine/cry and NOT shoutiness as you do this exercise. This will help you to start leaning on the right kind of muscle co-ordinations to achieve your belt.
Spending a considerable amount of time on these skills, your body would have gotten used to those sensations. Try to tap into that sensation when you sing high notes instead of shouting.
Remember you've got to learn to sing light and hooty, first; whiny and nasal, second; in that order for you to be able to achieve a beautiful belt without tension or shouting.
Obviously, there's only so much I can guide you through on text format and without actually diagnosing your voice in specific and its troubles like I do with my students. But I've tried to make the answer as universal as possible so that more singers can gain benefit from it. Please feel free to use the terms I've highlighted and explore on Youtube for more exercises. But here's a disclaimer, ANY pushing at this point is bad. Make sure that whatever you do, you learn to avoid tension in the throat. It's important to remember that there IS a better way!
Hope that helps :)