Experience matters. There's only so much you can learn at school. I'd prefer an engineer who has already solved real-world problems to someone who's exceptionally good at solving whiteboard puzzles anytime.
Would it be sufficient to have solved many different types of problems encountered while working on side projects?
One thing to keep in mind though is that part and parcel of solving engineering problems at scale is working and communicating with others. If you've been working on these side projects all by yourself then that experience might not be as valuable as if you had worked on these problems in a team. This certainly is possible with side projects, too of course, like when you have an open source project with several contributors or a side project with several remote developers. It's just that the term 'side project' usually implies a one-man-band doing everything by him- or herself.
Good employers value knowledge and skills, not experience.
Apply to well-known tech companies with lengthy interview process: they don't care much about your past.