I have really skewed RNA-seq data, what's the best way to normalise it? Preferably in R! My data frame compares the RNA-seq reads from many genes in different tissues.
The reads look as follows.

I tried using log to make it better but still looks pretty skewed.

Are there any better transformations/functions for this in R?
This is part of an assignment I have, I have used in the past edge software and agree it will do a better job but in this work I'm not meant to use any of the pipelines.
Welcome to RNA-seq. This is not at all unusual, as RNA-seq raw data are not normally distributed. People typically approximate it via the Negative Binomial distribution. I suggest you check existing packages as Devon suggests in his comment for normalization and differential expression (or anything else). Typical choices are DESeq2, edgeR or limma, but there are others as well.
This guide might help you get started. Please also use google to explore existing threads both here at SE and over at biostars.org.
Most basic questions have already been discussed there and are informative to get started.