Is it just my lack of training and experience, or is it very difficult to read the intervals between the two clefs? Especially intervals of more than an octave like 10ths and 12ths. Can someone share a tip or way to learn these intervals more easily. Perhaps some kind of method for counting the intervals that can help me learn them quicker?
On the assumption you recognise and name each of the two notes involved:-
Let's take C below middle C, and E on the first line, treble clef. Imagine the top note dropped an octave, interval is M3. Add 7, to take the top note back to its original place, M10.
Same lower note, but top note is G♯. Drop G♯ down an octave, interval is aug.5. Add that magic 7, taking the top note back to original, interval is aug,12.
One more: same low note, but top note is B, middle line treble clef. Drop that one octave, the interval is, temporarily, M7. Add the magic number, making the initial interval M14.
All this, as you seem to know intervals up to the octave, according to the question.
Learn the the scales and chords at first. Then you can derive the intervals from the chords and the note names. From their function you can immediately identify which interval it is:
Adding intervals is different from adding numbers. Like two seconds are a third, a fourth and a third are a sixth, so you always have to count one less than in arithmetic addition: when adding an octave you add 7 (like Tim says).