Every system of rhythmic instruction comes down to two basic things: some form of counting system, and some form of "learned intuition" or "muscle memory".
When we first start to learn rhythm, we might be banging a stick on a block or clapping our hands...1...2...3...4. After a while, our brain starts to remember the pattern and we don't really need to count anymore. But when we encounter a new rhythm, we don't have any "stored" memory of that new pattern. As you get better, you'll build up a "library" of stored patterns, so that you can just glance at a few bars of rhythm and you'll be able to spit it out without so much as a thought. It becomes a series of stored patterns in your brain that no longer require conscious thought. (think - do you have to consciously measure the speed and distance and timing to get a spoon to your mouth? - when you were a baby, you did, but now it's automatic). But it takes time!
So your question comes down more to: what do you do RIGHT NOW that might help you on your rhythmic journey. A few people have pointed to subdivision. That's super important and a MUST if you want to understand what's happening in between notes. Subdivision is simply tapping or thinking about each note in double time or quadruple time to help keep your notes even or accurate. You can use numbers or sounds when counting. "ta" for quarter notes, "ti ti" for eights and "tiri tiri" for sixteenth notes can be helpful for some. You can clap your hands on the quarter notes and say the subdivision. So you might be clapping 1, 2, 3, 4, but say Ti ti, ti ti, ti ti, ti ti along with each clap or tap. Saying "one AND two AND three AND four AND" (where and represents the other half of the quarter note) is a good way of subdividing with just your voice. You can help yourself of course by clapping the quarters. It might feel like patting your head and rubbing your tummy at first, but this isn't a "child's game". Anyone of any skill level can benefit by practicing and changing up how they learn and rehearse rhythm.
An important thing to improve rhythm is to simply practice...with a metronome. As a start, try putting the metronome (or metronome app) to a reasonable tempo - say 60 beats per minute (60bpm). Clap along with the pulse or click. Then clap double-time (two claps per click). Then return to clapping with the click. Now switch it up. Say "ta" with each click, repeat for twenty to thirty clicks, followed by double-time "ta" with each click. (to change how you store the pattern - clapping and speaking are two different ways of storing the information - related - but still different!) If you're an instrumentalist, then try the same thing on the instrument. Now combine two ways of representing the pulse...clap and say "ta" or play and tap your foot or hand (along with the metronome). Then go into subdivision on one of your patterns, the clap or the voice - then switch it up.
Brain studies have shown that pattern retention is significantly improved when a "guide pattern" is used (the metronome) (, as opposed to just doing it yourself from the page. (Neuropsychology: Lewis et al, 2004) You can move from that to triplets or to quadruplets and to dotted rhythms. You can find many examples online - a quick search shows some free materials at practicesightreading.com. You shouldn't need to buy anything....pick up an old hymn book at a flea market and practice singing or clapping the rhythms (with a metronome!)
A very important thing to practice is consistent tempo (which is why we use the metronome) This is overlooked by so many people and one of the reasons why rhythm can be sloppy. We tend to sing or play in OUR version of the pulse (or tempo) instead of THE ACTUAL tempo. This can lead to rushing and a poorly learned rhythmic selection because it fails as soon as someone else is "steering the ship" - or changing the pulse when they deliberately speed up or slow down (read: conductor or drummer). Practicing with that metronome eliminates your need to track tempo as well and you can focus on getting the relative timing accurate.
One more thing that often gets overlooked: rhythm and tempo are very 'internal' things to us. But we are required to synchronize those pulses with other players, pianist, conductor,etc. The biggest way we synchronize as classical musicians is with the conductor or principal player (baton waving or head nodding+breathing together, respectively). These things are often visual cues....but we've been practicing our rhythms with auditory cues!! (Click click click....) So....incorporate into your practice some use of the metronome with a flashing lightand try and stay synchronized. This will significantly solidify your pattern learning.
Jazz musicians may "feel" the music but they all still count. They might just be counting groupings of learned patterns now instead of individual beats - and all in the subconscious.
Practice practice practice (and occasionally turn off the metronome, too, because you can't take that up on stage with you).