install bowtie2 from sources cannot find -ltbb
You need to give the path to the source files for the compiler and the full path to the compiled libraries (all parent directories included) for the linker.  You also need to use either CPATH or CPLUS_INCLUDE_PATH instead of C_INCLUDE_PATH to point to the oneTBB source, since bowtie2 is C++ and C_INCLUDE_PATH will only have an effect for C (see Environment Variables Affecting GCC in the GCC documentation for more info.)
For example like this, inside bowtie2-2.4.1, without making any lasting changes to the shell's setup in this case:
$ ONETBB=$HOME/soft/oneTBB
$ export LIBRARY_PATH=$ONETBB/build/linux_intel64_gcc_cc5.4.0_libc2.23_kernel4.4.0_release
$ export LD_LIBRARY_PATH=$LIBRARY_PATH
$ export CPATH=$ONETBB/include
$ make

After it's built you'll still need the library path to oneTBB defined (via LD_LIBRARY_PATH) so bowtie2 can find it at run-time.
But if you don't have a specific reason to compile from source I'd recommend avoiding it.  You could go with what h.mon suggested and use an existing binary, or use a package manger to install it like zorbax suggested.  Conda has a package for bowtie2 in the bioconda channel, up through and beyond version 2.4.1, and you can use that as a package manager without root access.