We're looking to run a fast level-pace game DnD4E like game, and would rather not bother messing with new or house rules. Is there/can you recommend a similar, D20 based, game, where levelling up happens quicker (say, on the order of 1 level up per session)?
Yes, The system you're looking for is 4th edition with the house rule of "level up every session".
This is actually a remarkably popular house rule and it doesn't impact 4e adversely at all. I would, personally, recommend every other session just to allow people to be used to a level (when I levelled every session, the changes (especially in epic) were... pretty severe).
Mind you, this produces a remarkable amount of paperwork and many players without system mastery will simply check out of the process. The likely outcome is that the players with maximal system mastery will carry the others.
If you want rapid advancement, you may want to choose an alternate system with simpler mechanics. However, that is outside the scope of this answer. From my experience, levelling up in 4e is significantly simpler than 3.5. An option for both systems, however, is to plan out the entire character's development at the start.
It looks like you've found your answer already but I thought I'd recommend 13th Age anyway. It is d20-based and is similar in many ways to D&D4 as it was designed in part by one of the latter game's designers.
It is probably easier -- in terms of the paperwork involved -- to upgrade your 13th Age character with each level and the rulebook does give specific advice on how to run a campaign in which the characters gain a level each session.
All that said, 13th Age has a condensed level spread so a 30th level D&D4 character is roughly equivalent to a 10th level 13th Age character. That may be a little too rapid!