On a few easy songs like "Wonderful World" I can play, but not on something like "Turkish March." What are suggestions to help me to play the bass part better?
Most pianists are overly right hand-oriented. They don't relaize that the left hand is the conductor, the leader, the "main man." The left hand generates rhythm, harmony, phrase, everything! When the left hand knows its stuff, the right hand feels like a soloist who is well-supported by her orchestra.
But normaly what we see is the right hand trying to carry a deficient left hand, and struggling to do it! So really investigate your left hand part on its own, and make it an entity unto itself, and only THEN bring your right hand back into the picture.
Try "Dolly's Dreaming and Awakening" by Osten, it's a simple song, for the bass part, and it helps you get rhythm and everything down well, it was the first song I learned double-handed.
The reason why most people have trouble playing left-hand parts in most pieces is because many are right-handed. Now left-handed people will obviously be stronger on the left-hand. Now why does dominance relate to playing? Because obviously people who are right-handed has better strength in muscles, coordination, control, etc. And left-hand is kinda used to "hanging there"...being kinda lazy.
Well. Don't just keep playing pieces with the lazy left-hand...The only way to make it at least the equivalent of the right hand is to work the left hand out. For an example, scales. Don't try to do "easy fingerings" for the left hand, do the hard ones so you can really build up the weaker muscles. You should also try pushing and pushing yourself to play faster and faster with stronger muscles adding legato and dynamics in the scales. And when starting new pieces, start with the left-hand, listen consciously to the left-hand and practice the left-hand part more often. You must constantly push yourself ..."play evenly! use correct volume. is left-hand playing the melody right now? what are certain notes in the left-hand that should be heard? am I etc am I etc"... Focusing also learning the purpose of the left-hand part and also building up and setting goals to improve both technically and musically will probably automatically make you better. Even just the correct mindset will help.
But for now, keep working on the left-hand technially. Try using Czerny's technical works... like Czerny 100 or 125...theyhave lots of etudes/studies/whatever you'd like to call them that focuses on the muscular strength and focus on the left-hand.