I'm not sure if this is a glitch or not, but it's the second time that the guards in Solitude attacked me for no apparent reason out of the blue. The strange thing is, in both cases, I was able to enter the city normally, but as soon as I turned in a quest they attacked me.

I'm only lvl 15 and I took the lighthouse quest so I'm not sure if that has anything to do with it. I was able to get around the issue by running away to another city, wait 36 hours then coming back to Solitude and paying my bounty. Is this normal or is it a bug?

You must have done something to earn a bounty and piss them off, normally however that means you will be "arrested" outside of the city as well... So it sounds like a bug matched with a completely normal occurance.