This tends to happen at the end of every expansion, where the people that have been playing longer just want to coast through and set higher and higher standards to get into their groups. It doesn't help that Garrosh on normal is a long fight, so people don't like to wipe on him multiple times. Unfortunately, your options are a bit limited:
Join a guild
The best option is to join a raiding guild that actually raids weekly. Note that SoO has been out for close to a year at this point, so many raiding guilds are experiencing fatigue. It becomes harder and harder to retain a consistent crew, especially as WoD beta invites begin to open up.
Something you can do is to look at guilds you think you might be interested in and see if they are actually inside the raid at the times they claim they raid. So, if a guild says they raid Tuesdays and Wednesdays, /who their guild on Tuesday night and see how many people are online and raiding. You won't be able to distinguish between LFR, Flex, or Normal, but if less than ten people are online, they may struggle to raid normal mode weekly. (Or, maybe they fill in with consistent out-of-guild members.)
Join a cross-realm group
It sounds like you're already trying this and not having much luck. There are addons like oQueue which can help with cross-realm raiding and may let you find some other group that doesn't require a Garrosh kill.
You can also try pleading by showing off your ilvl and your actual normal mode/flex mode experience. I don't recall significant mechanical differences between flex and normal aside from tighter dps/healing requirements, but it's been a long time since I've done Garrosh on flex. Maybe you'll get lucky and someone will waive the achievement requirement for you just to get their run going, especially if you demonstrate knowledge of the mechanics.
Or, you can try to cheat the requirement with an achievement faker or by linking someone else's achievement in the hopes nobody reads the name. However, if you do this, make sure you're prepared for the fight in advance, or else the raid leader will figure it out and be unhappy.
Lead your own raid
I'm sure you're not the only one in this position. You could try forming a group yourself (thus, setting your own requirements) and attempting to find people that are willing to join. Be prepared to spend a while setting up the group, and these groups usually are only good for one night.
If you don't need gear, extend and repeat until you work your way to Garrosh. Some people won't mind skipping the first half of the instance for a shot at the BoA weapons and tier token off Garrosh.