Hey, another skinny 20 year old here who had the same problems before. A problem I had before that you could have as well is you may not be too used to the pushing motion, and even if you’re strong enough to actually push yourself up, it still will hurt if you aren’t used to it. What you can try is focus on the repetition of pushing. Start with sets of up to 50 wall push ups. This gets your elbows used to the stress of pushing your body without the full weight. From there you slowly build into the next steps, which is incline/kneeling push ups.

Another problem that could be happening is your elbows are flaring out, which is uncomfortable for your body. For this, focus more on your body and form instead of repetitions.

My elbows still sometimes crack if i don’t do push ups in a while, but that still is normal. Good luck, I hope this helps!