I realized that when the Spotify iOS app is low on disk-space, it will cease to download new songs.
You might want to check by going to iPhone Settings -> general -> usage to see the disk space. Spotify needs at least 1GB of free space to operate.
If it still lags, try doing a clean reinstallation by logging out, removing the app entirely and installing it again from App Store.
There could be old caches stuck in the app.
You should have min 1GB of free space for streaming and if you need to sync playlists, the rough estimate would be 100 songs to 1.4GB.
Furthermore, you will see an icon spinning when syncing and those in a grey download icon beside the track name means its pending sync, and those which are successfully synced will be turned green.
If you have to sync about 1,400 songs, it will take at least 3GB to 6GB of disk space on your iPhone.. Do you have enough space? (You can check for free space via the Settings app -> General -> Usage)
I have synced about 1.4K plus songs and it is using about 4.3GB.
Also, each Premium account has a 10,000 offline song limit, spread across 3 devices. This means each device can store 3,333 songs max for offline access. (So once you have the 4th device on available offline mode, it will kick out one of your device.)
If you are on a mobile connection you should go to Settings -> Sound quallity -> Mark Sync with mobile connection (last option).