Faction and navy cruisers will get you a little farther, but even then, there are some L3 missions that may be challenging. I definitely wouldn’t attempt L4 missions with a cruiser. That said, there are some L4 missions I wouldn’t try to do with a battlecruiser, either. (I recall my first major loss many years ago was a Drake in a particular L4 mission).
The thing to remember is that there’s more to the training angle than just the hull. Training the minimums, you can quickly get into the larger hulls. But without the support and weapons skills also trained up, you’ll find yourself losing ISK by going too big too quickly.
Keep in mind as well that L4 missions are a different breed than L3 and below. L1-L3 are mainly scaling numbers. L4s tend to add little tricks or quirks, like specific enemies that when shot pull the entire pocket. There is a website with mission reports that are often helpful for L4s (they have reports for lower level missions, too).
Knowing what you’re going into and carefully balancing your damage and resists against your mission opponents will get you farther with smaller hulls.