Is this sentence correct?

'at school' would be better. 'In school' implies that the students keeps their bikes inside the school buildings.

Students are advised not to ride their bikes to school because of a shortage of space.

Yes. Correct in American English. But a bit slang.

"...there aren't enough places to ....". Not many. This is better.

"Students can't go to school by bike because there aren't many places to keep their bikes in school." should have the word "many" replaced with "enough" and "in school" with "there": "Students can't go to school by bike because there aren't enough places to keep their bikes there."

There are always other ways to express an idea, but your sentence as it stands is perfectly correct.

Whether it's TRUE or not is a totally separate issue.

Generally correct as a informal way but formal way is different. 

It's correct but not precise. After all, students can go to school by bike, even if there are not many places to keep their bikes. Instead, they choose not to go to school by bike. (The word "keep" is okay, but "store" is a little more precise.) 

More precise would be, A lot of students do not go to school by bike because there aren't many places to store their bikes at school. 

...there aren't any bicycle racks to park their bikes. or

...there aren't any bicycle racks.

Yes, but it would be even better if it started with 'Most'

Your sentence is not, grammatically speaking, 'incorrect', but it would be better if you wrote '...there aren't any places to keep their bikes...' rather than '...many places'.