Read from one csv file and write to different multiple csv files depending on content

What happens when you do this? I shuffled your code around to make it generic, so that you can give any list of cols and it will make 1 output file per col, and look up the col in a dictionary of csv writers.

import csv
output_path = r'C:myfolderlarge_file.csv'


def get_fhandle(col_name):
  return open(
    os.path.join(os.path.dirname(output_path), f"column1_{col_name}.csv"),
    "w", encoding="utf-8", newline=''
  )

cols = ["01", "02", "03", "04", "05"]
files = {col: get_fhandle(col) for col in cols}
try:
  writers = {
    col: csv.writer(file, delimiter=";", quoting=csv.QUOTE_MINIMAL)
    for col, file in files.items()
  }
  with open(output_path, encoding="utf-8") as in_f:
    cr = csv.reader(in_f, delimiter=";")
    header = next(cr)
    for writer in writers.values():
      writer.writerow(header)
    for line in cr:
      col = line[0][:2]
      if col not in writers:
        raise ValueError(
          f"'{col}' is not a known column. Should be: {cols.join(', ')}"
        )
      writers[col].writerow(line)
finally:
  for file in files.values():
    file.close()

The use of contextlib.ExitStack() allows one to clean things up substantially, particularly if we set up a little dictionary to map what keys in our input map to what output file we want to write to.

import contextlib
import csv

files_out = {
    "01": "column1_01.csv",
    "02": "column1_02.csv",
    "03": "column1_03.csv",
    "04": "column1_04.csv",
}

with contextlib.ExitStack() as stack:
    ## ---------------------
    ## Set up a bunch of csv writers
    ## ---------------------
    writers = {
        key: csv.writer(stack.enter_context(open(value, "w", newline="")))
        for key, value
        in files_out.items()
    }
    ## ---------------------

    file_in = stack.enter_context(open("small.csv"))
    rows = csv.reader(file_in, delimiter=";")

    ## ---------------------
    ## write a header to each output file
    ## ---------------------
    headers = next(rows)
    for writer in writers.values():
        writer.writerow(headers)
    ## ---------------------

    ## ---------------------
    ## write rows to the appropriate file
    ## ---------------------
    for row in rows:
        key = row[0]
        if not key in writers:
            print(f"no file defined for key {key}")
            continue
        writers[key].writerow(row)
    ## ---------------------

To answer your specific question of why your code does not currently work, let’s look at:

with open(output_path, encoding="utf-8") as in_f:
    cr = csv.reader(in_f, delimiter=";")
    cw01.writerow(next(cr))
    cw02.writerow(next(cr))
    cw03.writerow(next(cr))
    cw04.writerow(next(cr))

    for line in cr:
        if (line[0][:2] =="01"): cw01.writerow(line)  
        if (line[0][:2] =="02"): cw02.writerow(line)  
        if (line[0][:2] =="03"): cw03.writerow(line)  
        if (line[0][:2] =="04"): cw04.writerow(line)  

here:

cw01.writerow(next(cr))

essentially writes the first row (the header) of your input file to the first output file. We then repeat this writing successive rows of input to the outputs.

After writing the first 4 lines of input to one line in each of the four outputs we then read the remaining lines in the input file (the last two in the example) and write them to files 3 and 4 as appropriate

So, your specific solution would be to use next() once to get the header from your input file and then write that to each of your output files:

with open(output_path, encoding="utf-8") as in_f:
    cr = csv.reader(in_f, delimiter=";")
    headers = next(cr)

    cw01.writerow(headers)
    cw02.writerow(headers)
    cw03.writerow(headers)
    cw04.writerow(headers)

    for line in cr:
        if (line[0][:2] =="01"): cw01.writerow(line)  
        if (line[0][:2] =="02"): cw02.writerow(line)  
        if (line[0][:2] =="03"): cw03.writerow(line)  
        if (line[0][:2] =="04"): cw04.writerow(line)