I wrote exceptional verbs and not irregular verbs because the verbs we’re about to learn are not really irregular. They just suffer a small change in the indefinite conjugation. I’m talking about:

VERBS IN -S, -SZ, -Z

The difficulty arises in the 2nd PS form because the usual suffix is -sz. It would be quite difficult to pronounce such words, so here is the solution:

Verbs in -s, -sz, -z take the suffixes -OL, -EL, -ÖL in 2nd PS indefinite conjugation!

Examples: lesni (to peep), mosni (to wash), nézni (to watch), rázni (to shake), tenni (to put), mászni (to climb)

Deep verbs 2nd PS indef.: mosol, mászol, rázol
High vebs 2nd PS indef.: lesel, teszel, nézel

Note that mászni is actually an -ik verb: mászom, mászol, mászik…

-IK VERBS

The -ik verbs end with the -ik suffix in 3rd PS indefinite conjugation (hence the name). These verbs have more special features:

-They take -m in 1st PS indefinite conjugation instead of -k.
-They take -ol, -el, -öl in 2nd PS indefinite conjugation just as those verbs in -s, -sz, -z do.

Examples: mosakszik (to wash), esik (to fall), öltözik (to dress up)

1st PS: mosakszom, esem, öltözöm
2nd PS: mosakszol, esel, öltözöl
3rd PS: mosakszik, esik, öltözik

The rest of the conjugation is regular.

Note that there are real and false -ik verbs. Real-ik verbs end in -s, -sz, -z: mosakszik, esik, fázik

Example for false -ik verb: tűnik (to seem, to appear). It ends in -n, so it’s a false -ik verb. You conjugate such verbs regularly: tűnök, tűnsz, tűnik…